At what altitude does a wing become useless?
Maybe wings never mattered...

Altitude:

The answer depends a bit on the design of the craft in terms of control surfaces/wings. Atmospheric reentry begins at the Karman line at an altitude of 100 km (62.14 mi / ~ 54 nautical mi) (327,360 feet) above the surface. Which is generally when air resistance starts to matter but that does not mean control surfaces will do anything at precisely that altitude. In terms of practical vehicles that have been flow:  

The space shuttle starts maneuvering with its aero surfaces ~30 miles up 


  At an altitude of approximately 30 miles [158,400 feet/50km], the orbiter makes a series
  of maneuvers and S-turns to slow its speed. At 9.5 miles in altitude
  and at a speed of Mach 1, the orbiter can be steered using its rudder.
  The on-board computers fly the orbiter until it goes subsonic (slower
  than the speed of sound: Mach 1). This happens about 4 minutes before
  landing.


Neil Armstrong famously flew the X-15 high enough to need the reaction control rockets to keep from skimming off the atmosphere, he was up around 207,500 feet. 

So somewhere up around ~175,000 feet, practical control surfaces start to work. Its possible aircraft could be built to get above this region but they would need the ability to either fly up there or survive the decent which is a whole different can of worms. 

Speed:
Any speed so long as your thrust to weight ration is greater than 1. All rockets start from a speed relative to ground of zero. Rockets simply have engines capable of providing 100% of the thrust needed to move (generally directed in the up direction). Planes have enough thrust to overcome drag then use the wings to help them with the lift (up direction) portion of the problem. There are aircraft that have a thrust to weight ratio greater than 1 and are capable of "flying" in a way without wings.