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Formations and detachments confuse me, especially when you throw allies into the mix. (I am a returning player from before McKinsey starting taking care of organizational restructuring in the grim future). This discussion assumes I'm using a battle-forged army.

I want to run Imperial knights as my primary detachment and ally with some SM. Maybe play some smaller fights with SM as my primary. I also plan to throw some Inquisition in there and Officio Assassinorum, but the latter two seem rather straight forward.

Primary
1) As a primary detachment can I take an Imperial Knights "Baronial Court" formation?
2) Do I need to first choose a Household Detachment before taking formations?
3) Assuming I take 5 knights in the Household Detachment, can I organize 3 of them into a "Baronial Court" and the final 2 are in no formation, just a part of the detachment?

Allies
4) As allies to my Imperial Knights, can I take an SM "Strike Force Ultra" formation?
5) Or do I need to take a Combined Arms Detachment of SM first and THEN add formations? (First a libby and 2 scout units as CAD, then add a "Strike Force Ultra" as a formation)
6) If I take a CAD, do I organize the CAD units into formations? (a CAD of libby and 3 scout squads, with the 3 scouts organized into a "10th Company Task Force" formation)

Further allies
7) Can I have more allies? (Imperial Knight primary detachment, SM allies, Inquisition allies and Officio Assassinorum allies)

I hope you guys can help clear this up for me.

A formation IS a detachment.

Yes. For the reason peregrine stated.

1 Yes
2 No. An army composed entirely of formations is fine as formations are a special type of detachment
3 No. A unit cannot belong to two detachments so it cannot be part of the household detachment and the baronial court.
4 Yes
5 No, see 2
6 No, see 3
7 Yes, as many as you like. There is no limit on the number of detachments and every one could be from any faction.

Thanks a lot for this guys, this really clears things up for me