This is a great learning experience for you! It's not going to be cost effective, however. Finding bad capacitors is difficult. Just guessing doesn't cut it, so often people will just replace a bunch of them in hopes of solving a problem. Often, it doesn't, but places will sell whole kits of replacement capacitors for some products. One of the best ways of testing electrolytic caps is with an ESR (effective/equivalent series resistance) tester, which start at about $100.
http://www.anatekcorp.com/testequipment/capwizrd.htm
Electrolytic caps often fail from heat and age anyway, but your chances of fixing random failures by randomly replacing a few capacitors is, IMO, low.
BTW, the solder bulbs aren't so hot, but the spring loaded sucker is much better. Technique counts, as you are discovering. Using braid tends to heat the area up too much, causing damage. If you do this all the time, you need an electric vacuum desoldering tool, $500-1000.
AFAIK, the Apple display board is single layer. I hope you read the CRT warning FAQ at repairfaq.