New 03 Impala few questions (engine, transmission, and wheels)

[tr10av]

Just bought an 03 Impala about a month ago and its a pretty nice car. big and smooth ride which is what i bought it for. Just have begun to notice a few things and was looking for some help/opiions:

1) Sometimes I'll start the car and when I put the car in Reverse or Drive the transmission won't engage. I either have to turn the car off/on a couple times to get it to engage or if I press on the gas and raise the RPMS it will click in. Theres no jerk when it engages but you can tell that it definitely bumps a little.

2) When my gas is low (1/4 tank of less) it becomes difficult to start. I was thinking possibly a fuel pump problem when it has to work extra hard to get the gas out but I'm not sure. Otherwise it starts instantly with enough gas in the tank.

3) What are the biggest size tires I can fit on the car without rubbing or scraping. I was thinking about putting on some 22s. And if I did this would I have to upgrade brakes/suspension at all?

Sorry for the long post but I appreicate any insight you guys can give.