Taking on parties much larger than yours is really the only way. I suggest just garrisoning your army somewhere and going around with just your companions. It's more risky, but yields much higher rewards (more renown, more loot, more money, more exp for you and companions).
I don't even hire any troops until well into the game, and just go around with my companions until then, fighting bandit parties many times larger (mainly sea raiders). That way I can accumulate 1000+ renown relatively quickly. It's difficult at first, and you will lose sometimes, but eventually your companions can far surpass top-tier units, able to steamroll parties 10 times their size. Of course, the above strategy won't work if you make your companions into purely party-skill specialists. I allocate each companion just 1 (or 2 at most) party skills so that they don't have to sacrifice too much of their combat ability.
The Relation link I've looked at. I'm slowly building friends but DAMN it takes a long time! Like I said, the only way I've found to do it at all is to beat them on the battle field while a mercenary for another faction, then let them go. Doesn't half piss off the guys I'm working for though!
If you are doing well in a war, release everyone you capture to accumulate honor. You will automatically get 1 relation point for every 3 honor points with all honorable lords of the realm, of which there are 20-ish. So if, for example, you got 120 honor, you'll automatically get 40 relations with all honorable lords even if you never met them, and they'll be listed as your "friends".
c) How do I take a castle to store troops without ending up in a permanent state of war with the faction I took it from? I just want to take it, then make friends again before they take it back...

and will it help? I think I need higher "right to rule" right? At what point of renown should I be sending out my companions to spread my fame?
Well if you take their castle, you bet they'll fight you, and likely they won't want peace for quite a while. Also, there's a very good chance that another faction will also declare war on you immediately afterwards, since you're a weakling upstart with low right to rule, no lords and 1 castle. Prepare for a long and bitter war afterwards. When I took my first castle in the current game, I had Swadians and Rhodoks pummeling me for what seemed like forever. They fought tooth and nail for that damn castle, and attempted to raid my village like every other hour. It was hard.
You can send out your companions at any point, regardless of renown.