How much damage does an asteroid-and-ship collision do?

The Stellar Phenomena or Terrain section of the vehicles and starships chapter of the CRBs. Flying/driving through obstacles or other hazardous or tricky terrain requires a Piloting action. The difficulty is based on current Speed and half the vehicle's Silhouette size, rounded up. Don't forget the vehicle's Handling! Zero to three Setback dice can be added for the density of the obstacles, hazards, or trickies. This can and should change from round to round as you move through more and less dense pockets.

At top Speeds, a stock YT-1300 has a difficulty of PRR- (Speed 3, Silhouette 4, -1 Handling), and a TIE/LN has a difficulty of PPPRR+++ (Speed 5, Silhouette 3, +3 Handling).

From round to round in the asteroid field, the YT's Piloting difficulty can look like anything from PRR- to PRR----. (I hope that pilot has taken up some Skilled Jockey ranks.)

From round to round in the asteroid field, the TIE's Piloting difficulty can look like anything from PPPRR+++ to PPPRR+++---.

If the Piloting check fails, the current Speed is lowered by 1. Threats and Despairs can be spent to indicate actual collisions.

A TIE fighter has only the one person in the cockpit. The pilot cannot Piloting and Gunnery in the same turn.

Oh, but wait a minute... Maybe that's another question. Is the Piloting check due to terrain/phenomena take effect as the pilot's maneuver, or the action? It's basically hazardous movement and movement comes from maneuvers. In the SW CRBs, it seems like it's a maneuver - you're just having to make a Piloting check for your maneuver. However, in the Genesys CRB, it's an action called Dangerous Driving. Maybe it's both? Or either?

How many Threats do you spend for collisions with the obstacles, and how much damage do they do?

According to the "Stellar Phenomena or Terrain" part of the Core books, the Piloting check stems from maneuvering within difficult terrain. For actions, you'd use the difficulty given in the action's description and add Setbacks according to the Navigation Hazard chart.

Stay on Target has spending charts for specific kinds of difficult terrain. Collisions may come through 2 Threats (make the same check as for moving through difficult terrain or suffer a minor collision) or Despair (minor collision and decrease speed by 1 if you still pass the check, major one + the "Knocked off course" crit if you fail).