The problem with the PS2 probably stems from trying to display a PAL signal on an NTSC tv. The easiest fix is to find a PAL -> NTSC converter, like this:
Note, I’ve never used this unit before nor done business with that company, it’s just an illustration of the device you need. Your existing PS2 games probably wouldn’t play on a US PS2, so it isn’t as simple as finding a domestic console.
The DVD’s have 2 problems, the first is the same PAL issue as above, the second is the DVD region is wrong for a domestic DVD player (US is region 1, UK is region 2). There are a few solutions for this. You could get either a region 2 or region-free PAL dvd player, and use the same converter as above. You could rip the dvd’s on a computer, convert PAL->NTSC, and burn new region 1 dvd’s for a domestic player/tv. Or you could try to find a PAL tv and PAL region 2 dvd player - but it may be very hard to find ones that run on 120v, 60Hz power.