Congrats!! Looks like you found a nice gem of a car. I agree with the others, the problems on this vehicle are minor and can be expected at this age.
I would wait until the battery dies to replace, and do have the dealership charge it up. Costco Kirkland batteries are reasonaly priced, have a good CCA rating too. My brother has one on, bought in 2001 or 2002. It is a very easy do it yourself job, and I did this in an emergency situation on a road trip. Just disconnect the old battery, reconnect the new. Always disconnect ground (negative) last, and reconnect ground first. All you'll probably need for tools are pliers and maybe a screwdriver. Be careful not to short anything and electrocute yourself, and be careful of battery acid.
Rad flush is definitely worth it, especially if maintenance records show it was some time ago.
Tranny flush also a very good idea.
Just how bad are the rear tires? If possible, replace them as a full set of 4. Just 2 only is acceptable, but try to match to the other 2 as closely as possible. The tire forum guys can help you out.
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Also look into a winter tire/wheel package, but seeing as you helped your mom out on this recently, I'm sure it's old news to you. You might be able to hold out on this til next season, seeing it seems like spring is here in Vancouver now (though I did eat my words one March the day after I took my winter tires off).
Right rear strut will probably be okay for a while, but expect a replacement a ways down the road.
For the $350 that the dealer wants to do the tranny, coolant and alignment sounds not too bad. I would let them do it as well.
Depending on what you can substantiate from the service records, new spark plugs and an oil change could be in order.
Happy motoring, enjoy your new ride!!!
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