Can US-Philippine dual citizens vote in both countries and run for public office?
Voting in elections
Can US-Philippine dual citizens vote in both countries?
Most likely, yes. Both countries have universal suffrage meaning that there is (from Wikipedia)
the right to vote of all adult citizens, regardless of wealth, income, gender, social status, race, or ethnicity, subject only to minor exceptions.
I have not found any such exceptions applying to this situation.
Regarding the Philippines it says:
Males who were over 25 years old and could speak English or Spanish, with property and tax restrictions, were previously allowed to vote as early as 1907; universal male suffrage became a constitutional right in 1935. Women's suffrage was approved in a plebiscite in 1937.
Regarding the United States it says:
In 1971, the 26th Amendment ratified, which granted suffrage for men and women aged 18.
And section 1 of the 26th amendment reads:
The right of citizens of the United States, who are eighteen years of age or older, to vote shall not be denied or abridged by the United States or by any State on account of age.
Dual citizens and public office in the US
Can US-Philippine dual citizens run for public office?
This depends on the public office in question as there may be restrictions.
For example, to run for President of the United States, one must be a natural born citizen of the US, according to article 2 of the US Constitution:
No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States.
The rules for US Congresspeople seem less strict. For example, US Representative Omar naturalised to the US from Somalia:
In 2018, she was elected to the U.S. House of Representatives, marking a number of historic electoral firsts: she is the first Somali-American, the first naturalized citizen from Africa (...) to serve in Congress.
Dual citizens and public office in the Philippines
In the Philippines the case is clearer, from inquirer.net:
MANILA, Philippines—Filipinos with dual citizenship must first give up their foreign citizenship under oath before they can run for any elective post.
The Supreme Court issued the ruling on Friday as it upheld the Commission on Elections’ unseating of the Vice Mayor Teodora Sobejana-Condon of Caba, La Union for failing to renounce her Australian citizenship under oath as required by Republic Act No. 9225 or the Citizenship Retention and Re-Acquisition Act of 2003.
“Failure to renounce foreign citizenship in accordance with the exact tenor of Section 5 (2) of RA 9225 renders a dual citizen ineligible to run for and thus hold any elective public office,” the court en banc said in a 24-page decision penned by Justice Bienvenido Reyes.