1) Yes, on a crosswind takeoff you put aileron into the wind and use
rudder to stay aligned with the runway. One instructor had me do an
exercise in which I lifted up on one wheel in a crosswind takeoff and
proceeded a good distance down the runway in that condition.

2) Airliners bank into the wind too, but the degree of bank is much
less, since the crosswind is a smaller percentage of airspeed. I'm sure
engine strikes have happened, but it's not a common problem.

3) In gusty winds, it's often necessary to make sudden corrections of
both the rudder and aileron to stay centered on the runway. Many years
ago I had the wind change from a right to a left crosswind at the
instant of touchdown. Slammed me into the runway on the nosewheel and
right main. Thought for sure I had broken the airplane.
--
Gene Seibel
Confessions of a Pilot - http://pad39a.com/publishing/
Because I fly, I envy no one.