1. Assignment from an HQ to a division and back is manual. It's costs one AP per assignment.
2. Yes, it's sitting at the Army level or possibly higher. If the HQ or higher level up HQ is in range you can assign an SU directly to the division.
3. It starts historical or as close as they could guess. Some HQs have tons of SUs and others 0.
4. It's a choice. You could attach tanks to infantry corp HQs for some armor support OR to motorized divisions for some armor punch.
5. SUs assigned to divisons always are involved in combat. SUs assigned to HQs within 5 hexes of a combat unit may be included providing the HQ didn't move during the turn. It must pass a series of tests to be included. If the HQ moved, the SU may be involved only if the combat unit does a deliberate attack and then must pass some tests to be included.
6. Yes. I haven't checked, but my guess is that an HQ in range of a combat unit with flak units assigned might have a chance to be included as well.
You'll have to play around with auto-assignment. It's tricky. Setting all your corps to 3, for example, will tell the system to attempt to get 3 of each type of SU out there. Since there isn't 3 of everything, nothing will move. What I do is set my corps, first turn, to 1, my armies to 2 or 3, army groups to 3 and high command to 3. This will automatically transfer some of the corp HQs that start with 20 up to the army level (or higher) where they can be reassigned. Reassignment still costs 1 AP, so transferring down from army group to army, army to corp, or anything to a division will cost. But it does save a bit on AP points.
At some point, I start locking the HQs so nothing automatically moves. I like the motorized battalions, flak companies, panzers, stugs assigned to my panzer group divisions. One army (the 18) and the Panzer groups get the pioneers and nice arty. THen I start trying to get 3 arty into all my other corp HQs so they have some punch.
You'll see that some players start reassigning SUs from mobile divisions back to corp HQs when winter comes. So Stugs, Flames, Jags get moved to corp HQs for defense support.