If my glass jar of pasta sauce does not have any special "use within x number of days after opening" warnings, then is it okay to keep it after opening, but still refrigerated, until expiration date?

No: No, I would use the common sense rule of however long you might keep leftovers. Use the smell and look test, too. It is probably good for a couple of days. Definitely not for weeks.

Depends: Most ready-to-use sauces bought from grocery shelf, if kept refrigerated, will get moldy before growing significant amounts of bacteria. They usually have preservatives to prevent spoilage. Usually if it looks & smells as it did when 1st opened, it is safe to eat. However, it's generally best to eat these within 2 weeks of opening. The expiration date refers to unopened items.