I'd say that while you could abbreviate season 1, omitting it entirely probably isn't a good idea. But you could cut it down to 8-10 episodes (depending on how many of Chekhov's guns you want to see hung, vs. just fired) without doing noticeable harm to the story.
And, if you want Chekhov himself to be there as a gun is ung, you have to watch "Mind War."

"The Quality of Mercy" should probably be watched, because when its A-plot is referenced later on in the series, it's almost incomprehensible without having seen the episode. I have an incredible soft spot for "The Parliament of Dreams," too, and "Deathwalker" gets a fairly important callback.
My problem with "The Quality of Mercy" is that it's a good episode, but not a great one. However, it hangs too many of Chekhov's guns to skip. The way I figure it, the episode was written to (a) hang the guns and (b) provide a break between "A Voice in the Wilderness" (both parts)/"Babylon Squared" and "Chrysalis," as the previous three episodes were very "heavy" and the season finale is, without being spoilerish, a bit of a game changer.
"Believers," however, is just so incredibly ham-fisted. It's got all the subtlety of a sledgehammer slamming into your skull, as though the script was originally titled "Baby's First Ethical Dilemma" or something.
I completely agree. "Belivers" has always struck me as a story that would have been more at home on TNG than Babylon 5. The scene between Sinclair and Franklin in the Rock garden in particular. I was SEEING Sinclair and Franklin, but I was HEARING Picard and Crusher.
Though I don't picture Crusher in Franklin's spot. I could, conceivably see Pulaski being there. However, I could also see some deus ex machina somehow taking the story in a different direction than B5 went in.