Like everything involving chaos, and to a lesser extent the Imperium and 40k in general, theres about a zillion different interpretations.
Plus, we don't always get to see the intricacies of the Chaos command setup beyond some charismatic Demagogue being the guy in charge, with a few minions, and the rest of them are mooks.
Gaunts Ghosts gives the best look at an organised, highly structured Opposing Force. The chaotic forces of the Sabbat Worlds basically ape Imperial command structure, even before Gaur becomes Archon, with powerful Cultist leaders taking the positions of Lord Generals, subordinate to the Archon and his court.
The Cultist leaders forces ranged from screaming nutcases like the guys Nokad the smiling had, mass-conscript zealots with IG style organisation and equipment like the Zoicans, to guys like the Blood Pact and the Sons of Sek, who might give the Cadian shock a run for their money, with well connected chains of command, officers, NCO equivalents etc. I.e. Mr General speaks to his command staff, who then speak to the commanders of smaller formations, etc etc.
Can for example any of the CSM Codex characters show up in a warzone and say i have the favour of (insert chaos god here) and all your base are belong to me?
If their balls are big enough, and they've got the muscle, then sure. Abaddon for example, despite being a bit of a tard, has got the muscle. Fabius Bile would probably turn up and offer his services, while fucking about for his own reasons, and god knows what Ahriman would do.
How about a Daemonhost which is possesed by a high ranking demon doing the same?
High ranking daemons don't tend to have the same set of goals as more mortal characters, a high ranking daemon might turn up and wreak havoc, or co-opt the nearest group for its own gribbly ends.
What are the Daemon Primarchs doing and why didn't Peturabo step in to stop them killing each other?
Navel gazing is the best way to put it I've seen. Mortarion is lounging around on a world he's transformed into the image of his former homeworld, leaving Typhus as the most senior remaining member of the Deathguard we know about. Angron was the last one to actively go out and kick ass, but GW haven't done anything with him since.
Fuck knows what Alpharius/Omegon are doing, given their tendency for super secret bullfuckery. Fulgrim is probably jerking off, Peturabo is being depressed, Lorgar is probably giving regular masses and scribbling more lunatic prose, and Magnus is likely scheming to get more of his dudes back/murder some Space puppies.
Chaos marine organisation ranges again from the fucking deranged, to the heavily regimented. A rule of thumb is that it vaguely resembles that of loyalist marines, with champions subbing in for the Sergeants and other officers.