How far can you run an electrical cord, that is plugged into a 20 amp circuit, and still run a circular saw? If the saw starts slowly, will that harm the saw?

I don't think you'll damage the saw, but I'd think if it starts slow then your cord is heating up big time.
What's the amps on the saw - and the length and gauge of the extension cord?

If you are just going to whack a few boards off, it may be OK. I would be more concerned about the saw heating up than the cord. If it is outside laying on the ground, it isn't going to set anything on fire.

Using a heavier cord could make a difference. I have a couple of 25' 14 gauge cords. I also have a couple hundred foot of 16 gauge cords, and haven't had a problem. My circular saw is a light duty one.