Let me first say that there is
a lot of room for handling cohorts and followers in different ways, depending on the campaign setting, on the preferred gamestyle, and even on practical matters such as how many PCs and how many cohorts are at the table.
There is no one best or one-size-fits-all way to handle these. For instance, I prefer to RP cohorts and followers myself as the DM, but other DMs want the player to RP the cohort just like it was a second PC (I have my reasons for this, but it's not the point of the thread).
Here are my average opinions on the matter, but even I would handle things differently from time to time.
Cohort and followers serve the leader. They willingly put their lives in his hands, because they see something in him, they believe in him. They are loyal towards him.
That's a good starting point to keep in mind. When I present the Leadership feat to players, I always remember to point out that they are in fact spending a feat in exchange for cohort's loyalty. I know that there are DM's who disagree, and let the cohort betray the leader, but I usually think that if there is no guarantee of loyalty, why spending a feat and not just get a NPC ally through RP?
There is some general difference between cohorts and followers, but I was a bit confused between the two while reading your post... let me know if I mix them up. For reference, I believe that the cohort (usually one, tho some DMs allow to take Leadership multiple times) you get from Leadership is going to follow you in dungeons, quests, combat etc. essentially becoming an additional member of the party of PC, while followers will attend "offline" businesses and be therefore of marginal utility (although once again there is freedom here to handle things differently).
But what attitude could followers have towards each other? I'm talking here about those followers that do actually work together or at least know each other (I know that a character could possibly nab himself followers from two rivaling organizations or races and they would obviously not like each other). Are they likely to be friends?
I simply would assume an average "friendly terms", like two people playing in the same team or working in the same company. Since followers don't actually "follow" the PC in the dungeon, the vast majority of the times you can just go with "hide the details" i.e. there is usually no need to tell if Bob and Rob had a fight while preparing your horse barding and that caused a delay. Personally I'd just treat all followers
as a whole. As such, I would still give the guarantee that
as a whole, the followers are doing their job, by getting along fairly well. This means for instance, I wouldn't say one day that "you lost half your followers because one of them was a traitor and poisoned their food".
Of course, things like that can become story hooks if you want, but in general I would advise against using these idea to purposefully diminish the effectiveness of having followers in general (i.e. to punish a player who dared thought the Leadership feat was going to be useful).
If you had 2 cohorts instead, I would do the same: on the RP level, they might anything from best pals to openly hating each other, as long as this doesn't change the outcome for the player.
What about cohorts/followers by proxy? How could they feel about the leader of their leader?
Assuming the cohort shares at least some (and preferably many) aspects with the person they're all following (for example the same alignment, organization, religion, outlook/opinions, skills, abilities or background), what attitude could followers of his leader have towards him? He's basically something like a lieutenant. Would that make them at least on friendly terms with him, if not also loyal and subservient?
This is more tricky... not sure if I understand right, are you asking what is the relationship between the cohort and the followers of the same PC?
I suggest to just treat the cohort and the followers more or less equally, i.e. expect the followers to treat the cohort as one of them, except that he is generally a levelled NPC so he probably gets more respect. He might have some duties over organizing or leading the followers into their activities, but this is not mandatory.
If the leader is a PC, what could his cohort and followers feel about the other PCs?
As a rule of thumb, I'd say the cohort is effectively an ally NPC that follows the party in their adventures, therefore he has (on average) the same relationship with the other PCs as he would have if he was a full-regular PC. He doesn't have absolute loyalty to other PCs like he has for "his" PC, but then he should have enough loyalty to follow the same "good party rules" that the DM has set for all players (which in my case, it means no attempts at damaging another PC's or her properties).
The followers instead, I'd probably have them mostly ignore the other PCs, unless their Leader specifically instruct them otherwise. That's because anyway IMHO the purpose of the followers is giving some minor support outside adventures.
More mechanical question - how reasonable would it be if I gave Leadership to the cohort and allow him and his leader to basically "share" the same followers? I'm asking because there are some feats and abilities that give benefits to followers. Not too many, but there are. So instead of using up feats of the main guy, his cohort would be the designated "inspiration provider".
I've never thought of that, it's an interesting idea...
I think this is OK. In strictly mechanical terms, the cohort would get his own cohort and followers. But this looks like it would be something to avoid, at least because the cohort's cohort following the PC party will be of much lower level hence mostly useless, but would still complicate combat. OTOH, if you do like you suggested, there is no "exploding number of followers" and no additional cohort. In a way, it sounds like you're giving up something, but I think this would be fine. You'll still get some benefit, and you're still spending a cohort's feats for that, so the benefit is not free.