I'm not interested in the official Israeli view, which is Jerusalem.
Do other countries have an official stance on which city they recognize as Israel's capital?
As a side question, does the Palestinian Authority recognize Jerusalem as Israel's capital?
It's not at all clear that countries even have official recognition of other countries' designated capitals; all that really happens is that a place for the embassy is chosen. One of the first tools of diplomacy is not saying what you don't have to (the second being not saying much of what you do have to) so an announcement that "we're moving our embassy from DC to Pittsburgh" probably ends there, with no further mention of why. A lot of popular media will call this something like "de facto" recognition of the capital; more careful outlets will say things like "Israel controversially designates Jerusalem as its capital but most countries have their embassies in Tel Aviv".
So on Wikipedia's List of diplomatic missions in Israel, only the US and Guatemala have embassies in Jerusalem. All the other 86 are listed in the greater Tel Aviv area. Paraguay announced they were moving and then backtracked. Here in Canada the current-opposition Conservatives have occasionally floated the idea of moving Canada's embassy, but they were in power for ages and didn't do it then.
Basically, it's a big political hot potato.
Of a number of secondary missions in Jerusalem, some of them are "accredited" for service only to the Jerusalem area, and these countries' Tel Aviv embassies seem not to serve that area. Many countries have consulates and trade missions that serve only a designated region of the host nation, or conversely include service to a neighbouring country, so it's not a definite statement that "we don't think this area is part of Israel" ... but it's not not saying that.
Yes, most certainly.
The UN (though, not a country, a good starting example) officially doesn't recognize Jerusalem as Israel's capital:
The United Nations General Assembly does not recognize Israel's proclamation of Jerusalem as the capital of Israel, which is, for example, reflected in the wording of General Assembly Resolution 63/30 of 2009 which states that "any actions taken by Israel, the occupying Power, to impose its laws, jurisdiction and administration on the Holy City of Jerusalem are illegal and therefore null and void and have no validity whatsoever, and calls upon Israel to cease all such illegal and unilateral measures."[55]
And China, for example, does not (same source):
China recognizes East Jerusalem as the capital of Palestine.[60]
On the other hand, some countries do recognize Jerusalem as the capital of Israel. The United States under President Trump controversially did this as recently as the end of 2017:
The warnings come as President Trump considers moving the U.S. embassy from Tel Aviv to Jerusalem, effectively recognizing the ancient city as Israel's capital despite Palestinian claims to East Jerusalem. The move could come as early as Wednesday, according to media reports.
So, it varies widely by country.
As for a comprehensive list of which countries have which stand, I couldn't find one. But know that a large number of countries don't, as shown by these 128 countries that voted against the United States' decision to do so.
As for the Palestine Authority, they recognize Jerusalem as their capital, not Israel's, hence why there is a lot of conflict:
Earlier this year, the Palestine Liberation Organisation's (PLO) Central Council, the second-highest Palestinian decision-making body, has recommended revoking recognition of Israel until the latter recognises the State of Palestine in its 1967 borders, with East Jerusalem as its capital.
Something I forgot to address earlier: if countries don't recognize Jerusalem as the capital, I believe your title asks what city would they be recognizing?
This is very unclear from everything I've seen, but it appears to be Tel Aviv (emphasis mine):
On December 6, 2017, US President Donald Trump announced the United States recognition of Jerusalem as the capital of Israel2 and ordered the planning of the relocation of the U.S. Embassy in Israel from Tel Aviv to Jerusalem.2
This suggests to me that the other city would be Tel Aviv.
[sic]. As for the other portions of my wording you edited, I didn't see how it made anything more clear, so I have rolled it back.
[2][4]) to be incorrectly interpreted as link formatting (e.g. in your last quote, there are two links labeled "2" that point to two different articles - and they're not the sources referenced in the Wikipedia article under those footnotes).
Most countries recognise the 1967 borders set by the UN.
https://en.wikipedia.org/wiki/United_Nations_Security_Council_Resolution_242
The resolution doesn't specify a capital and I therefore the closest you will find is where the embassies lie. This is also why the movement of the US embassy from Tel Aviv to Jerusalem in 2018 was controversial.
Russia and Australia view Jerusalem as Israel's capital. Most countries have no official position on this topic.
The embassies of Guatemala and the US are in Jerusalem.
The location of embassies is not a definitive statement on considering some city the capital. The US embassy of Bolivia is in La Paz, not the capital Sucre. (La Paz is considered the 'de facto capital, but that is begging the question.) Many countries hold a "non-resident" embassy to smaller countries in a third country, and that is of course not a statement on the capital of that smaller country.
