I think @kaizen is on the right track.  If it is a battery/voltage issue, that’s worth fixing.  There are some other things you also can do that are cheap to free, like cleaning the MAF sensor and throttle body.  And also look at Scotty’s videos over the years on fuel system cleaners.

In general, yes, an A3 is just not built well enough to be a forever car, and parts are $$$, so you don’t want to waste much more on it.  But they’re nice little cars, so if an inexpensive tune up will get you a few years out of it, great.  (And yeah, start saving for a replacement)

I am not sure how a mechanic or q more knowledgable person would do it. This is how I would go about it. 

I would tackle the P3XXX and the P0562 first. 

With P0562, it has to do with the voltage. Which could mean a bad battery or a bad alternator. If you can get a good compatible battery to test with, or you know how to run an alternator test, that may be a good route. 

If this is solved, many of the other codes may*** possibly be solved.  Not sure though.

With the P3XXX, assuming it is it too hard to access, I would examine the spark plugs and the coils.

A battery, spark plugs and coils are part of normal maintenance. But the question is if that will solve most of your problems. 

I'd try to fix it. Heck, the price was right and you've been driving it and it doesn't run all that bad so it's probably nothing too serious.

You're getting it scanned for codes. You need to buy your own (inexpensive) scanner.

Then erase all those codes and now that you have a new battery and have run that 5 year old fuel out of it see which codes come back.

5-year-old gas is a recipe for your Check Engine Light coming on. The only time I ever drove on 5-year-old gas was a '79 Catalina with a carburetor, right after I bought it. Almost anything that's flammable could be theorically used. Not with a computer that monitors everything. 

Change the spark plugs check, or change the air filter, basically do a full tune-up. Run it low enough to trip your gas light, and fill up immediately. That should help solve your problems. 

From a financial viewpoint, I'm not sure that fixing it up makes a lot of sense.   Audi's of the age and mileage yours is tend to be money pits even without the complications of sitting for so long.  Something will probably always be going wrong, and it will never be cheap.

Check your battery, voltage regulator. Check for corrosion and burning on the main fuse block. If the fuse block/sam burned up or has corrosion everything that runs to it or is connected can have failure or faults if effected. While your at it also look at the main fuse and relay.  Having experience working on and owning a German car it sounds like some type of voltage issue.

Another possibility is the Ecu/ PCM is bad. That also can explain all your faults. .. Now Should you keep the car. It depends on many factors but if it's just the battery or something minor then why not. As long as there is nothing major wrong. German cars cost more to maintain and fix then others especially if you can't do everything yourself. You need to take everything into account 

Why was it sitting for five years?! What were the problems it was having?

Is it really only worth $300? 

I will take note of all this advice and take a crack at it. Thanks everyone