3-4 no by Training Grounds I mean the benefit it gives to Retinue Size not the benefit it gives to Levies.1-2. No. Retinue doesn't belong to the levy and is personal/bound to the characters' title only.
3-4 It increases the vassals levy, thus your max. potential of levy to raise from that vassal. So yes.One vassals' building doesn't change that much though, well depends on the usual rules.
5. Don't remember if family palace belongs to the levy. I believe not, but not sure at all, sorry.
Other source:
Regarding levies in general: http://www.ckiiwiki.com/Levy
Wiki said:Training Grounds in a Castle give 20 cap per level.
http://www.ckiiwiki.com/Retinues