1. Do they go on the initiative count round as their "master"s"? Or do the roll for initiative separately.
2. If they work for a PC, does the PC control their actions or are they controlled by the dm.
1) According to RAW, they get their own initiative.
2) Generally, no. Animal companions with intelligences of 2 or less must be ordered around with the Animal Handling system, cohorts are DM controlled by generally loyal and take orders.
Both are RAW, but my playgroup generally ignores 1) and lets the player control Animal Companions. Planar Allies/Bindings and Cohorts are generally going to be more powerful so the DM may want to control them in combat- at the very least they have the ability to override your commands. A player can't order their cohort to an obvious death without the DM going "well would Jimbo the Bard Friend really be okay with that?"
If you're in an argument with someone, I'd say that it's my opinion that cohorts/companions/etc acting on their player's initiative is generally just better, even if it's not in the rules. DM decides everything after all.
1. Do they go on the initiative count round as their "master"s"? Or do the roll for initiative separately.
2. If they work for a PC, does the PC control their actions or are they controlled by the dm.
1. I have always seen this played as going on the player's initiative. This reduces the number of initiatives that need to be kept track of during combat, and simplifying combat is something GMs like.
2. a) Cohorts and Animal Companions are class features (the leadership feat being something a PC takes as they level). As such, they should be controlled by the PC. I have usually seen these as run by yhe PC but with GM override if the action is unreasonable. Rarely done with a well thought out action. When I have had such, I sometimes ask the GM if my cohort is willing to do something rather than assume so on a borderline action.
2. b) Planar Allies and Planar Bindings are the results of spells that bring others to help the PC, and as such the PCs do not control their actions. GMs run these characters, but may allow you to run them if they trust you not to abuse the privileged. If your command is in line with your contract, they will likely do as you ask. The further afield you ask, the less likely they will comply unless it is in their writeup's stated self interest. So asking a fire elemental to burn something not related to your contract will likely work since they live to burn stuff. Asking a Lyraken Azata to fight when not in the contract is much less going to happen.
/cevah