I will update with references when I have a chance, but I am currently writing from memory. The short answer is that there isn't much good polling of these populations on the issues of concern to you, in part, because both of these populations are fairly small, and both of these populations are highly fragmented internally into sub-populations whose views are not necessarily cohesive.
Musings on the immigrant African's place in American politics without statistics can be found in a 2007 Op-Ed piece in the Los Angeles Times.
There are an estimated 2.88 million Afro-Caribbean people in the United States out of 37.7 million African-Americans overall as of 2010. But, this population is divided amongst many national immigration stories. Some survey data on Afro-Caribbean attitudes on some issues can be found here and in the study it reviews.
Cuban immigrants, some of whom are considered black, tend to vote Republican and are more conservative. This is in part because Cuban immigrants in the U.S. tend to be people who fled a left wing regime in Cuba as political refugees, many from around the time that Castro's regime was put in place. There is quite a bit of polling data to support this conclusion. About 85% of Cuban immigrants and about 64% of Cubans in Cuba consider themselves to be white. There are about 180,000 black Cubans in the United States.
Puerto Rican immigrants, some of whom are black, tend to vote Democratic and to be left of central politically. This is particularly clear in greater New York City where Puerto Rican migration to the U.S. is concentrated. There are about 4.6 million Puerto Ricans in the mainland U.S. of whom about half identify as black (8.7%) or of mixed race.
Dominican Republic immigrants, most of whom are black, tend to vote Democratic and to be left of center politically. I am not aware of specific issues upon which they differ from African-Americans. Many Dominican immigrants are also concentrated in Great New York City, although there may also be other areas of regional concentration. There are about 1.4 million Dominicans in the United States, of whom 30% identify as white, 13% identify as black, and the remainder identify as some other race or mixed race. Many Dominican immigrants in the U.S. are dual citizens who are participate politically in the Dominican Republic.
There are about 830,000 Haitian Americans in the United States, many of whom live in Florida. Many are black. They tend to favor Democrats.
Other racially African people from other parts of the Caribbean tend to migrate in current or colonial powers associated with those countries in Europe (e.g. the U.K., France, and the Netherlands). So, there has been relatively little immigration from these countries in the Caribbean to the U.S., and I am not aware of polling analyzing their political tendencies or views as a distinct sub-group. Studies on the voting patterns of these populations in Europe would probably provide insight into U.S. voting patterns.
There hasn't been a lot of migration from Africa to the U.S., mostly because other countries (e.g. France in West Africa, Central Africa, and Northwest Africa; the U.K. in Southern Africa, Nigeria, Egypt, Sudan and East Africa; Italy in Libya and Ethiopia; Portugal in Mozambique) have historical colonial relationships with those countries and provided more natural targets for migrants. The only country in Africa with a historical colonial-type relationship to the U.S. is Liberia, and to the best of my knowledge that relationship has not given rise to significant enough migration to the U.S. to create a notable political community in the U.S.
In all Africans make up 3.3% of U.S. immigrants in recent times from more than a dozen major source countries and a variety of religious backgrounds. A very high percentage have college educations.
In 2009, 74.4 percent of the African-born population reported their
race as Black, either alone or in combination with another race.
African immigrants identified as Black at a much higher rate than the
native born (14.0 percent) and the foreign born overall (8.6 percent),
and accounted for 33.3 percent of all foreign-born Blacks and 2.7
percent the total Black population in the United States.
Racial self-identification varied widely by African country of origin.
For example, nearly all immigrants from Ghana (99.7 percent), Somalia
(99.3 percent), Cameroon (98.8 percent), Nigeria (98.7 percent), and
Ethiopia (98.2 percent) reported their race as Black, either alone or
in combination with another race, compared to 4.6 percent of
Algerians, 5.6 percent of Egyptians, 8.1 percent of Moroccans, 13.8
percent of South Africans, 56.7 percent of Tanzanians, and 65.7
percent of Cape Verdeans.
From here. More background on African immigrant populations in the U.S. can be found here. References to studies of economic and educational prospects of African immigrants can be found here.
Muslim immigrants (a significant sub-set of whom are Sub-Saharan African) tend to vote Democratic but not as reliably as African-Americans (who are 83% Democrats or Democratic party leaning v. 8% Republicans or Republican leaning), and Muslim immigrants have more conservative political views than African-Americans, on average. Some of the data is in the General Social Survey, and some of this is in relatively recent political polling. Polling more often distinguishes immigrant Muslims from native born Muslims, than it does sub-Saharan African immigrant Muslims from other immigrant Muslims.
The Atlantic Magazine has a lengthy discussion of the Muslim vote in 2016. Only 15% of American Muslims backed Trump. 27% of American Muslims backed Bernie Sanders in the primary. More complete polling data on American Muslims in 2016-2017 is available here. About 57% of African Muslims voted in 2016 (v. 64% of Arab Muslims and 75% of Asian Muslims). Muslims tend to support the Black Lives Matter movement. About 25% of American Muslims are black and a significant share of that percentage involves native born individuals rather than immigrants.
Another survey of the electoral and attitude positions of American Muslims on the eve of the 2016 election can be found here. Some of the key findings were:
72% of Muslim voters said they will vote to elect Hillary Clinton,
while 4% said they will vote for Donald Trump, 3% will vote for Jill
Stein, and 2% will vote for Gary Johnson.
44% of Muslim voters consider themselves moderate, while 25%
consider themselves liberal, and 11% consider themselves conservative.
The percentage of those who said they are closer to the Democratic
Party remained constant, from 66% in a similar poll taken in 2012, to
67% today, after having increased from 49% in a similar poll taken in
Affiliation with the Republican Party remained relatively steady, 6%
today, 9% in 2012 and 8% in 2008.
In 2006, 17% of Muslims were Republicans.
Only about 5% of Muslim voters, as opposed to Muslims generally, however, in the U.S., have African ancestry. Still, a Voice of America story in 2016 on the shift of Muslims to the Democratic party, would likely have applicability to many immigrant populations:
Once seen as a "natural" Republican constituency, Muslim-Americans are
increasingly leaning Democratic, and they are expected to vote in
record numbers for Democratic candidate Hillary Clinton in the U.S.
presidential election in November.
From overwhelmingly voting for George W. Bush in the 2000 election to
backing Clinton in the current cycle, the Muslim shift in political
allegiance has been precipitous, leading some critics to lament a lost
Republican opportunity to keep an increasingly influential voting
According to surveys conducted after the election, more than 70
percent of Muslims voted for Bush, and most of the 50,000 Muslim votes
in Florida went to the Republican candidate. . . .
If there was a high-water mark in the love affair between
Muslim-Americans and the Republican Party, it was the 2000 election.
In 2004, more than 90 percent of Muslim-Americans voted for John
Kerry; in 2008 and 2014, Muslims voted for Barack Obama, by 89 percent
and 85 percent, respectively, according to several estimates. . . .
In the 2011 Pew survey, 70 percent of Muslims in America described
themselves as Democrats or leaning Democratic, while 11 percent said
they were Republicans or leaning Republican. Those numbers have held
relatively steady since then. McCaw of CAIR cited another poll that
showed 55 percent of Muslim-Americans describe themselves as moderate,
while 26 percent identify as liberal. . . .
Many analysts thought conservative Muslims and Republicans shared
common views on issues such as homosexuality and the role of
government. The 2011 Pew survey showed that Muslim-Americans have
grown "considerably more accepting of homosexuality" since 2007. On
the role of government, the survey found that 68 percent of
Muslim-Americans preferred a bigger government providing more services
over a smaller government providing fewer services.
"So not only do they switch parties and now are voting Democratic, but
they're also adopting some of the policies and positions and ethics"
of the Democratic Party, McCaw said. "Traditionally, a number of
immigrants from the Middle East or South Asia are more socially
conservative, and there was a place for them in the Republican Party.
But I think as people grow and develop in America, [they] definitely
change their views and preferences over time; and more importantly,
their children grow up here and they might be voting different than
their parents previously had."
I haven't seen any hard data on the voting patterns of African immigrants who are not Muslim, mostly because the number of African immigrants who can vote is small, even in places like Minneapolis and Denver that have significant numbers of Christian African immigrants (e.g. Ethiopian Orthodox Christians), since relatively few are naturalized citizens.
Anecdotally, in my own dealings with the African immigrant community, they tend to be Democratic party leaning (since the Republican party hasn't been very welcoming to them), but are more conservative on some political issues than African Americans (e.g. issues related to gender and domestic violence and economics). But, my personal dealings with this community as a lawyer and local party officer are not scientific samples.
The Egyptian Christian immigrants that I have encountered (who are racially "North African" rather than sub-Saharan African) have been "swing voters" who are more conservative on many issues, especially social issues, than Democrats, but also suspicious of xenophobic attitudes in the Republican party and not as conservative as Southern whites, for example. Again, the numbers are too small to constitute even an informal statistical sample.
Studies on the voting patterns of these populations in Europe would probably provide insight into U.S. voting patterns. For example, immigrant Africans tend to be much more active participants in religion than native born people in the countries where they live.
A Voice of America piece notes concerns from an African immigrant political lobby about relationships between the U.S. and countries in Africa and about immigration law.
A piece of African immigrant Republicans from 2008 from PBS is here. It notes that Minnesota, with 60,000 to 90,000 African born residents is the fastest growing African immigrant population of any U.S. state and that this population leans heavily Democratic although it sent one delegate to the GOP convention in 2008.