The Close classification is really a points fudge to sort out an issue of over-costing if they were Loose ArmHrs. As cavalry with ArmHrs move, and suffer from terrain in the same was as Close it was a nice an easy solution.
As far as I know we have no idea about the formation spacing of Assyrian cavalry.
3 to a base is perfectly acceptable for Close formation cavalry, lots of players do that (me for one) and as long as you can see which is which on table there are no problems.
As for the non- Qurubuti cavalry in the Later Sargonid list I can't really comment on whether they should also have ArmHrs - but feel free to make a list amendment suggestion if you feel the evidence shows they should