How accurate can I expect an older watch to be?
I would expect a high quality mechanical watch in good condition to keep time to within a minute or two per week, some do better than this but you must consider that an error of one second per day is just one part in 86,400 and very few watches will approach this. Sometimes a watch will gain a little one day and loose the next, giving the impression of keeping 'perfect time'.

What is an automatic watch

This is a self-winding watch, it depends on the movement of the wrist. A centrally mounted rotating weight called a rotor winds the mainspring a little each time it is moved. It needs about 14 hours of reasonably active wear each day to keep the watch fully wound.  These watches can also be wound in the usual way if they are allowed to run down. Less active wearers should give the winding crown a few turns each day.

I have heard that you can overwind a watch, is this true?
No, manual wind watches need to be fully wound each day. You must simply wind it until you can feel the winding crown stop. Do not try to count the turns.

How often should a watch be serviced?
Most watch cases let in dust or tiny fibres which can interfere with the delicate mechanism; watches need to be dismantled and cleaned from time to time.  Watches need to be lubricated and even the most advanced modern lubricants dry out  after four or five years, although many watches will run quite happily without oil. A good guide is to have it serviced when the timekeeping deteriorates or the watch does not run a full day.

It says 'waterproof on the back, is it safe to immerse the watch?

Keeping a watch waterproof is very difficult, swimming in a waterproof watch is fine while it is under guarantee;  however keeping it waterproof involves regular replacement of case parts and seals and pressure testing. This is an expensive process to go through every couple of years to keep it fully waterproof and as the watch gets older it is not practical. My advice is not to swim in your watch or deliberately immerse it. 

Sometimes, particularly in a hot climate you may find that the glass mists over with condensation. You can probably cear this by warming the watch with a hair dryer, you can get it quite hot without damage. If the watch does not work properly, get it checked at the earliest opportunity.