Grats on purchasing the 5002, it's one of my favorites and definitely a classic.
Lug width on the faux PR version should be 22mm, you are correct. Pilot straps can be purchased at many places... I would look directly at the sponsors here on the board and see if you can get one custom made for your wrist size. HKTAN does good work but sometimes it takes a while. You can also look into Gorilla and Red12 straps... both of these strapmakers make really nice pilot style straps.
Regarding the deployant, are you sure it doesn't click down? Sometimes it might take a little bit of elbow grease to get the audible "click" of the deployant. I wouldn't be happy with a faulty deployant. If you bought it from a dealer you should complain to see if they will send you a replacement one that works.
For the adjustment of the PR, you're right. It's a relatively easy mod, you need to know how to take off the caseback, remove the crown and slightly nudge the PR needle with a toothpick to full. Wait a bit after moving it though, because sometimes it settles back down and will need another nudge to move it to where you want it. Be careful of the O-Ring when reassembling your case, good luck!