Some engines are fine from the factory, and for lapping on street tires, for 20 min sessions at a time, many won't have a problem. It would be prudent to google your specific car though. My poor old Neon would send coolant temps way up in 20 minutes and probably oil too, but I ran T6 5W40 and it seemed fine until it rusted out at 200k miles.
Going up a grade or two and synthetic is good idea IMO if you are running something that is commonly on a track.