Definitely leave it on always. It doesn't matter whether a computer is connected or not. The printer periodically does self-maintenance that's essential to keep it in good shape (especially the printheads).
And yes the startup time (from power off) can be quite lengthy. There are different types of startup it automatically uses, some are much longer than others. And all the clicks, whirs, wiggles, etc are expected..., after a while you get to learn what they actually mean..., and then you're One Of Us

lol
I can't comment on Mac but for PC/Win you don't def don't want to use whatever latest universal driver garbage HP has on their website now. They're not made specifically for the features on these printers, so many of the key things don't work. The bundle that came in the box is what you want to use to start. Up until recently the HP website still had some of the older versions (listed under older OSes) but it seems they've wiped it all now. So around here some of us have been resorting to passing drivers around.
I can only tell you some good driver/software versions for Win/PC, so hopefully someone else here can comment on the Mac versions (check other threads too).
Edit: ok you might be in luck, it looks like the HP website still lists legacy driver/software versions under older Mac OS versions. So you can browse the lists for a good fit. (For Win they recently replaced all legacy versions with the latest "universal" driver.)