Sadly, some mechanics do take advantage of young drivers. I've been a victim of that 20 some years ago, and that is why I became a mechanic specializing in water cooled VWs.
When ever possible ask for your old parts back, if they have a core (re-buildable item: Alternator, battery for lead recycling, etc), then you can't have the part back unless you pay the core fee.
Yes a stock battery is most likely the issue, it was a common replacement for us as well as the battery fuse box.
Alternators seem to be a more common replacement on the MKIV cars than the earlier MKIII. Bosch redesigned the alternator and they seem to last as long as the original factory warranty. MKIII alternators well into the 150K mile mark, I rebuild these as a hobby.
A new battery should meet the minimum CCA as the original, you can get a battery with higher CCA as long as it fits in the existing spot (assuming you have the original plastic cover. These covers seem to break or are missing.
Mice can enter a vehicle with a hole the size of a quarter. He may have climbed in through the rain tray drain hole. Then it's just a matter of gnawing on some rubber grommets and he's inside.
Those buggers gnaw on just about everything. I've seen some damage like this before over the years.
I won't scare you with my known MKIV issues, the list is quite long and its $$$$.
This really applies to any modern VW that has plastic parts on the engine, doors, seats, dash etc. Best to read up in the MKIV FAQs, purchase a Bentley manual, and some tools for the easy stuff.
If you have a friend who likes to wrench, have him/her help you learn to DIY. Many of the common repairs are easy, but some require special tools that would cost more than having a mechanic do it. If you can get affordable special tools and a willingness to DIY, you'll save big bucks.
That is how I got my start, out of need and lack of funds.
I'm on the west coast, so I can't really offer you advice on which shop to use.