I have wondered about piece movement statistics, for example:
I found an interesting page featuring the results of some data mining on a sample of 4,226,520 games, and from which answers to your questions can be extracted, at least for that large sample of games, which is probably reasonably reflective of what happens in general.
According to the numbers there, the average game has a ply count of 57.63 (i.e. total number of moves by both White and Black). So the number of times a single piece is moved is, on average,
57.63 / 32 = 1.8009
The page also lists the total number of moves played in these games by each type of piece of each color, from which we can derive the following per-game figures:
White piece moves
-----------------
P: 7.7708 total moves, .9714 per pawn
N: 5.1677 total moves, 2.5838 per knight
B: 4.6259 total moves, 2.3130 per bishop
R: 4.6985 total moves, 2.3493 per rook
Q: 3.5775
K: 3.1718
Black piece moves
-----------------
P: 7.7998 total moves, .9750 per pawn
N: 5.1576 total moves, 2.5788 per knight
B: 4.3511 total moves, 2.1756 per bishop
R: 4.5364 total moves, 2.2682 per rook
Q: 3.4869
K: 3.2875
These figures ignore the fact that some piece moves will have been made my promoted pieces, but on the whole such moves will be a very small piece of the pie, so these numbers shouldn't be too far off the mark.
As a minor modification to EDT's material, above:
According to the numbers there, the average game has a ply count of 57.63 (i.e. total number of moves by both White and Black). So the number of times a single piece is moved is, on average,
57.63 / 32 = 1.8009
A given ply will see only one player's piece moved (castling aside). This means that the simplest average to be obtained should approach:
57.63 / 16 = 3.601875
Note: this necessarily doesn't address any variables such as lost pieces, nor does it seek to address per-piece moves. This is intended only to address the special case of a "lossless" [in terms of pieces captured] game.
