Is it possible to create a new rivalry with a team in the game? If so, how would it work?

Not really, other than your personal rivalry with different managers.

There's been mention of teams from Blue Square getting to the Premiership and 10 years on still having rivalries with local teams and not creating any new ones.

i think this needs changing, if you have run ins with certain clubs ofter title challenges/play off place challenges/drawn together in cups a few times, plsu small rivalries if teams beat you to transfers etc.

I would have thought that if a team beat you and they got promoted/survived relegation at your expense, they should be your rivals.

Not if it only happened once. You'd have about 100 rivals if that were the case.

But if they beat YOUR team and you got relegated instead of them or they got promoted instead of you. Wouldn't it make a difference?

The way FM works is that the disliked/liked categories are gradually built up with a top score of 100. So basically if it happens once, it may go up by about 10, but it would then take 10 run ins with that particular team to actually cause them to appear as a rival.

The same thing happens when adding a player to the favourites, with different scores for different things e.g. giving them the captaincy earns 100, saying they are in good form is maybe 10 or something along those lines.

As Swarfega said, you couldn't just add to as results occur or in 40 years time you could have a page full of rivals.

I do agree that there should be some room for growth here. Rivalries are historical and don't happen over a couple of seasons, but I do think that a tussle over 5-10 years with a particular team should develop into a 'minor' rival and eventually major rival over a long campaign.

Not really.
You have your fierce local rivals, and you have the team you have built up rivalries with over the course of 10+ years, usually more.
Reading and Fulham were not rivals this season. Neither were Bristol City and Hull. Just teams with the same aim.

Well our(Brighton's) rivalry with the Palace scum was only over a few years, when Mullery was the gaff. He hated Palace. But then he went on to manage them!