Sounds like a perfect bike!! I bet it turns into something similar to the "dog" that folks find that are homeless and take it in. Maybe you should name it Lucky since it is now very lucky to have a home where it is wanted!!!
Probably a good idea to have the LBS give you a list of things you can work on. Would agree to have the wheels checked out but you could probably learn how to do the rest.
I agree with keeping the handlebars!! It's part of the make-up of the bike and you certainly would not want to do unneeded "plastic surgery" and change Lucky's character. The new bar tape will really, really make a difference. Plus we might just be getting you one step closer to a real road bike!!
Welcome to the world of downtube shifting!! It will make you appreciate STI or "brifters" even more over time!!!
What color is it?