Well not in the rival sense of a local derby or a deep ideological hatred but their was definitely a huuuuge competitive rivalry from around 2007-2012. The two teams just played eachother too often for it not be heated for those few years, but I think that microcosm of English football is very much over. Technically every team is a rival for they are all competing against one another, but you know what I mean.
Chelsea suffers through their locality, they barely play Fulham, QPR, and Brentford, so people fill in the blanks with Liverpool, Arsenal, Tottenham, West Ham, etc.
Well, John Terry I don't know about him really, he was an adulterer, did weird unsolicited tours of Stamford Bridge, and whatever happened between him and whatever Ferdinand it was; all seem a little weird in the sense that a lot of ex-teammates speak highly of him and he seems to have a great rapport with whatever fans stomach him. Seems like an average intelligence bloke that makes really stupid decision occasionally.
Every player has done weird shit, Terry's have been blown out of propotion, but even players that are generally loved like Gerrard (punched a bloke at a nightclub) and Lampard (was pissed at an airport and made fun of American tourists during 9/11, allegedly) have done it. Maybe it is the people I know, but nearly everyone has done something bad that they shouldn't have; even if they are good people.