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.