This is a longstanding Duolingo glitch. In order to be able to recognize the contraction when "have" is an auxiliary verb in a present perfect construction, it also allows it both when "have" is the main verb and when it's used as a modal.
Not really. "I got to" means "I was able to", unless you pronounce it "I gotta", which is way too colloquial for Duo. "I've got to" would be the grammatical version, and should okay, but might not be on the list of accepted answers yet.