Quickdry is pretty amazing stuff and will cover up all manner of sins, both surface quality (within reason) and any patchiness in colour that you are overcoating. You don't strictly need it, but some colours/tints definitely conceal what's underneath better than others. Better IMHO just to do it properly with an undercoat than end up doing a third or fourth top coat to get the finish you want.
No worries about overcoating acrylic over oil, so long as the oil is _thoroughly_ dry. I've just checked the label online and it claims 3 hours but I usually aim for next day to make sure.
Are you a decorator by chance? sounds like you've done this a few times!!