Can I move a piece protecting the king if it keeps protecting on its new location? Situation: there's a piece between the king and an opponent’s piece. If this piece were not there, it would be a check.

If I were to try and move the protecting piece somewhere and leave the king exposed, that move would be not allowed.

Could I move this protecting piece, even though when it's lifted the king is in check, but when the piece lands, the king is protected again?
Yes, that's legal.  For example:

[FEN "6k1/6r1/8/8/8/6R1/8/6K1 w - - 0 1"]
[SetUp "1"]
[CurrentPosition "6k1/6r1/8/8/8/8/6R1/6K1 b - - 1 1"]

1.Rg2


The rook can move along the g-file.  This is known as a "partial pin" - the rook is somewhat limited in its movement (and can't go to a3, for example) but it can still move if it still protects the king.