1) As you may know... = 아시겠지만 or 아는지 모르겠지만 2) 다음 표현을 배워 봤어요 는 맞는 문장입니다.

Hi Mojave, 1) First of all, "알지도 모르다시피 (=As you may know)" makes sense. I've heard of "알지도 모르다시피" quite often. However, I don't think I've heard of "볼지도 모르다시피" or "들을지도 모르다시피".. But they make sense too. If you use them, we will understand what you are trying to say, but it just sounds awkward. 2) Yes, '다음 표현을 배워 봤어요.' is perfectly correct sentence. Hope it helps.

to put it a nutshell, Heesob's comment is very excellent. 1) it does not make sense at all. as you already know , a langlage has a typical style in expressing something that is conformed by the language community. 2) good to use