I currently help out friends and relatives by fixing, cleaning, and
working on their computers in general.  My talents have become widely
known in this small town and I am getting more and more calls, from a
much wider area.  The monetary "gifts" I recieve for my help are not
enough for me to say that it is supplementing my income, but I am
worried about the legality of continuing this practice, without
reporting it as income.  However, I would prefer not to have to deal
with charging tax, registering a name, etc.

My question: Is it legal to continue without reporting it as income,
and if it is not, what route would I take to begin reporting it?  
Also, am I going to need a registered business name and an address for
the business alone?

I have been working in the industry as an independent consultant for
many years; I report my income  as a personal income as a self
employed.

When you are small the less exposure is actually better, meaning, not
having a business account, licenses and just operating as an
independent small consultant pretty much  takes away the
risk/liability you would have if you were operating more at a business
level. I wouldn't worry for now! As you grow you can strat doing
partial income report as  a self employed but make sure you consult an
account/tax advisor first to give you some heads up on the kind of
credits and deductions you can use; I would also try to do all the
monetary "gifts"  in cash :-), at least for now while you are still
small. Hopefully you will grow big and open a store/shop, start hiring
employees and all the fun associated with it; it is part of life and
it is fun. Good Luck!