cleanliness and salah

Salam,

yes.

if you didnt know, the underwear is a recent creation so im pretty sure the prophet didnt even wear underwear but just a covering from his navel to knee

see the Ahram?

Salam, yes, you can pray without wearing underwear. We know that it is must to cover our awra during Salat. And the cover (clothes) mustn't be transparent. The clothes must cover you from up and sides(under your clothes doesn't matter). You can wear a one piece clothes that covers your body and it would be better not to be VERY loose that shows your awra while Ruku3. Wa Allah a'lam

Yes its allowed

 Whether for salat or otherwise one must wear garments which cover one's satr (navel to the knees) effectively. for those who dress up in pants and shirts it is important - too tight pants are revealing aurah especially in ruku and sajud.