OK, here's the deal on 49er rivalries:
The Seahawks are a real rivalry to the 49ers. It is still rather new, and I don't think the peak all time hate is there like it is for other old foes, but it's getting there. It's certainly the bitterest division rivalry since the Rams in late '90s/early '00s.
Now as far as the Cowboys are concerned, I'd say over the context of the last 40+ years they would be #1. Knocked us out of two NFCCs in early '70s. We knocked them out in '81 NFCC. They knocked us out of '92 and '93 before we returned the favor in '94. All NFCCs and toughly contested. A lot of missed super bowls for both teams there.
But the thing is that the overall hate for the Cowboys never lasted very long past '96 or so, as they slowly faded from being a contender. Green Bay immediately took that mantle and we became their whipping boy for a very long time. Up until 2002 or so, big time hate for them.
However, again, hate for them faded as we dropped out of relevance. Watching them on any given Sunday I am more or less indifferent to their success (just like Dallas). I think I rooted against them a bit in late 2011 seeing that we might be playoff foes that year. We've won every matchup with them since then, so not a lot of reason to hate them.
New Orleans Saints are lodged right now between Green Bay and Seattle. Old NFC West hate is clearly still there, but more from them because they were irrelevant back then. Now that they had their time in the sun they feel a little entitled to still be considered a top 3 NFC team. I don't like them but reserve the hate unti we play them. Carolina is slowly entering that very same picture.
The NY Football Giants. Ok, that's a team I loathe. We've suffered a good amount of tough painful losses to them, and their fanbase seems so entitled to think they should hate us more for no other reason but perhaps our '80s dominance and knocking them out of a wildcard game they should have won in 2002. They twice KO'd Joe Montana in the playoffs, ended a 3peat and sent a few vets packing the next season, and of course denied our upstart Harbaugh team from the Super Bowl. I don't know if that's a debt that could be paid back.
And to top it off, as much as it can be contentious to talk to Seahawk fans on this board at times, it is nothing like interacting with Giants' fans on their boards. You would think they were in the NFC West splitting games each year instead of Seattle. Even when the 49ers were down for a decade, they still enjoyed seeing them suck. Outside of the NFC East they hate us more than anybody else in the league by a long shot.
I had "lost" my hate for the Giants when they were garbage in the '90s and mediocre up unti Eli's first super bowl win. I even rooted for them to beat the Patriots as they were the annoying team at the time.
Now after last season, I will vow to hate them till the end of time, rooting for them MAYBE if it were a elimination scenario to get SF in the playoffs. Last year that sorta was the case for home field advantage and I did initially want them to win the game. But being the ungrateful jerks those fans are to support them in that scenario has made me wish they go 0-16 every year.
So yeah this year, I'll root for the Hawks to blow them out....and for the rest of the NFC West to do the same to them. I want the Giants removed from any form of relevance, and we're almost there. Hoping Arizona hands them an L this weekend with their D playing like it's week 16 in Seattle.
Just to put that into context, 20 years from now I'll still hate the Giants winning SB 46 but will probably not be so bothered by the Seahawks winning SB 48.
As for the NFC West rivals, I rooted for Seattle and Arizona in their first super bowls but never the Rams. Might have been a different tune if SF was a legit team those years (and they were when the Rams played in them), but the Seahawk/49er rivalry only became REAL IMO in 2011, and took on another level in 2012 and has kept growing. It looks to stay on path for the next few years for sure.
What really drives the rivalry is the enemy fans hating your team, coaches, players, etc. Without hatred on a personal level, it's not a rivalry.
Just browse the forums and you'll see where the real rivalries exist.
Now that there's clearly no competition but Seattle, it's a very real rivalry.
ps - Any old rivalry can very, very quickly come back. Dallas, Green Bay, NYG, etc...