No. The auto focus routine requires you to be roughly in focus when it runs the first time and if you're roughly in focus you'll be good enough to do plate solving. You can add 'notes' that pop up when you start a sequence; I specifically have in mine to 'check focus' and 'slave the roof to the mount' to ensure I don't miss those steps prior to starting.
For your second question about being in balance and streaking images, you would find that guiding would solve your problem. Or, put in a proper settling delay (it's on the telescope tab) to ensure you don't have this problem. SGP assumes you're operating your equipment properly (it's balanced, you're in focus when you start).
The ability to set up targets based on altitudes has been brought up and it's in the hopper for inclusion at some point. As of right now, the only way to prevent something like this is to figure out when the object drops below your horizon (whatever defines that, trees, etc) and put in an appropriate time for that.
You can have SGP run a recovery mode (re-attempt centering/guiding every x minutes for x minutes). But, it wouldn't help in your case. It's mostly useful for the stray cloud that interrupts images.
I'd recommend really reading the manual a few times along with several of the tutorials (I've installed SGP... what now). They'll help you get the most out of your installation.
Good luck,
Chris