1) As you move faster you move through time slower when compared to someone observing you who is not moving as fast. They will mirror this and move through time more slowly compared to you.
2) When you move faster you don't notice time slowing at all. Only other people observing (stationary or slower thanyou) will notice anything. Distances may shrink though due to length contraction. This is an actual physics phenomena. Not an optical effect. So distances ARE shorter. Not SEEMINGLY shorter.
3) The laser DOES keep up with you because you always measure the speed of light relative to you to be the same. You'll see the man with the gun zooming away from you really fast. At some point the laser beam will be shot from that gun and will approach you at the speed of light, as expected. You'll see the laser beam comin at you at C, and the man moving away from you at nearly C (0.99c i think we decided on).