Is it possible to have lasik eye surgery to correct vision problems caused by an eye injury?

No. Most likely the eye injury will have damaged the eye enough that just doing lasik will not correct the vision enough, or sometimes the eye injury causes other problems that contraindicate getting lasik done. The best way to know if a good candidate for lasik is to have a preop eval.
Most likely no. Injury to the eye often causes decreased vision, corneal scarring, irregular astigmatism. These things are not correctable by laser eye surgery. See a corneal surgeon for an evaluation to be sure though.
Unlikely. Lasik will not correct serious problems from eye injuries. There is only a chance it may help if you can correct the vision with glasses despite the eye injury. It is very dependent on the type and location of the eye injury.
Unlikely. If he eye injury was severe enough to change the focus of the eye, you likely are not an ideal candidate for lasik. Sometimes, with limited scarring of the cornea prk can be used to clear the cornea - this is lasik without the flap performed directly on the anterior corneal surface.