1) This is too late for you, but www.toilettanklids.com has lids for virtually any toilet ever made! Prices range from aroung $90 to about $600. Yes, $600 for a LID. Your decision.
2) Regarding your problem, it is dificult to blame the new toilet completely for these symptoms. True a newer toilet may not flush the way your old one die, meaning it might not clear the bowl. But as far as affecting the main drain itself, the lowere amount of water in the flush, especially if you add up multiple flushes and the other fixtures on the line, there should not be a problem. I believe it would be worthwhile to get a camera in there to look for an improperly sloped line, bad joint, or other problem.
3) Cottonelle is my favorite paper for reasons we don't need to go into! However, it is the WORST flusher. I have a pressure assist toilet and sometimes would take 3 flushes on the cottonelle; Costco paper goes down in one flush every time. Sorry if this is too much information!