I have had HBC, Tantric's nintendo emulator channels, and WiiMC for several years now and was wondering if I can play games on wi-fi (mainly goldeneye 007 and brawl). I have heard from lots of people that if you try and use the wi-fi with either of these games after installing homebrew, it will brick your Wii. Please help!

The people who told you that are morons. No harm will come to your Wii playing online, I have done it numerous times, including in the titles you mentioned. The worst that can happen is you could get banned from online, but there are no reports of that ever happening to someone who hasn't cheated online. i.e. don't be a jerk and you'll be fine.

Does that mean that I can't use texture patches when I play brawl online?

I would certainly advise against it, but it may be OK. I believe (however, don't quote/believe me on this) that it is only "competitive" cheats (i.e. ones to give you an advantage over an opponent unfairly) that will get you banned. To be safe, I would advise against any type of cheat or hack while playing online.