1) Create a free account here, so you can answer questions in this thread, instead of scattering them all over the forum.
2) Does the check (for enabling wifi) stay checked, but the phone won't find wifi signals (or connect to one of them) or does it uncheck itself?
3) If it sees the signal and tries to connect, are you sure that's YOUR router's signal it's trying to connect to? If it's trying to connect to someone else's signal, and you don't know their password, it won't connect.
3) If it's your router, what kind of security are you using? (For example, Android won't connect if you're using RADIUS.)
4) You can factory reset the phone, but you'll lose all the data you've entered into it (including pictures, any other files, accounts, etc.) and it won't change what happens on wifi. Factory reset has nothing to do with unlocking (unless the phone is locked to the point that you can't use it at all, of course).