Welcome to the forum!
1) I would look for service records to see if there are repeat issues. Cooling system, valve cover gasket, looseness in the front suspension, typical BMW inspection items. Crash damage/repair history.
2) I would check for crisp shifts, with no flaring (slipping as the shift is made, indicated by rpm change). Check how smoothly shifts happen in a variety of conditions. You aren't trying to make it slip by abusing it, that will just cause the torque converter to mask any shifting problems. Drive it normally.
3) There are normal size car seats, and some very large ones. Normal car seats fit fine. With one carseat, you will probably have it in the centre position anyway.
4) See this 2004 product catalogue. Same dimensions for 2006, just some option differences.
http://www.bmwmotorsports.org/pdf/e5...%20Catalog.pdf
And for general E53 items, this folder:
Index of /pdf/e53
If I was to buy another E53, it would be a 2006, for the reasons you list.
Good luck