Is a queen vs rook endgame always winning? Other than specific scenarios where the side with the rook can draw, is it always possible to win as the side with the queen in rook and queen endgames?
The best resource I know for endgame statistics remains the ICGA endgame stats. The spreadsheet posted there contains pretty much all endgames up to 6 men. So for example KQKR is found in row 27 (assuming White has KQ), with the following percentages:

       wtm    btm   
W win 99.01 65.51
 draw  0.80  5.83
B win  0.19 28.65


So as long as White has the move, KQ is almost certain to win. Black victories with Black to move are likely due to be immediate capture or skewer. (Pin or fork of the queen leads at best to a draw, of course.)

Interesting to compare with KRBKR, row 107:

       wtm    btm   
W win 41.25  5.09   
 draw 58.70 94.06
B win  0.05  0.85


With White to move, Black does much better than in KQKR, which is curious since all that has happened is that the queen has been "unpacked" into its "constituent rook and bishop". With Black to move, the position is an almost certain draw: Black has lost the quick steals, but equally can protect himself from most of the situations where he might end up losing.

Note: (1) This is a tablebase based on computer analysis of all possible positions with up to 6 pieces. It does not refer to actual games played over the board. (2) The spreadsheet refers to "non-broken positions" which approximates to "legal positions".