Is there a way to get finer control over rotating lots when placing them? The arrow keys seem to move 30-40deg per click so it's often near impossible to get a lot lined up properly with the road.

I've noticed on occasion I couldn't use an exiting lot, but bulldozing it and replacing with a new lot, same size, almost always works. But recently removed a lot perfectly aligned with the road, the best I could do with the replacement was about a 20deg angle to the road.

Lot placement is noitoriously frustrating. I have never found out why the lots could be made to snap to the road sometimes and other times you can't get them parallel or close.

Did you try to press the alt-key while rotating the lot? They should rotate to any angle.

I tried with ALT key, seemed to have no effect. Does this need the testingcheats bit to work?

Yes testing cheats needs to be enabled and the cheat for using alt needs to be input.