To my knowledge, the only automatic movement that is going to be notably better in terms of accuracy is Seiko’s Spring Drive, which will get you to +/-15s per month. It’s actually a fascinating bit of mechanical and electrical engineering, and one of the few real innovations that have occurred in mechanical watchmaking in quite a long time.

In terms of adjustment: I usually set my watch a tad fast (i.e. round up to the nearest minute) on my seiko, because it doesn’t have hacking seconds and is kinda fiddly to get it to sync super precisely, and then just tweak it when I notice the time getting out of whack with what my phone or computer says.

Not often 😉 I have 2 automatics, 1 kinetic and 1 g-shock. The automatics wear ik most of the time 1 or 2 weeks. So I don't adjust them in that period. My Vostock is approx 1 minute ahead per week.