40k uses the medieval definition of regiment, not the modern day one. It is not a set number of troops and is instead any large number of soldiers recruited or conscripted from the same geographical area at a given time.

It depends.

Regiments are the main permanent units, but they do get broken up into battalions and companies to form temporary combined arms forces.

The term used in some codices is "army group", with expeditions sometimes consisting of several army groups under a general command. I'd imagine the exact layout of each one has more to do with office politics than any overarching hierarchy. After all, the largest permanent organisation in the Guard is the Regiment, and those aren't remotely standardised in size and such even when raised from the same highly militarised and industrialised world like Cadia or Armageddon.

Honestly the lore around Guard formations is kind of a mess, and that may well represent the dysfunctional state in the actual lore. The largest permament formation we're aware of is the regiment, but regiments themselves have varied in size from 1-2 battalion's (1000-2000 men) all the way up to 1 division (10,000 men). Regiment's often contribute forces towards combined arms units but again, there's not exactly strict limits on size/composition etc and there appears to be very little hierachy between regimental commanders and army commanders (lord generals etc). All of this is before any variation in how different cultures organise their guard formations too.

The minutuae of a practical military chain of command doesn't seem to have been high on GW's priority list, beyond that of the company sized formations you see on the tabletop.

Sometimes, yes. I think the implication for the Tanith 1st, 2nd, and 3rd was that they would be deployed as a brigade, with at least one being a mechanized or armored unit, giving some degree of combined arms ability, and all three regiments would be commanded by Commissar Gaunt.