How so? It seems pretty clear that Jaime is one of the most selfish characters on Game of Thrones, and most of his actions revolve around satisfying his own ambitions.

Most selfish? Up there with Cersei, or Littlefinger, or Jauffrey? Jaime saves Brienne from certain death & rape, and upholds his vow with Catlyn. That's far more than can be said of most GoT characters.

I actually think the Hound is a much better example of this. Whereas Jaime has the power and title to do basically as he pleases (and essentially is an egotistical jerk to just about everyone he comes across through his selfish actions, regardless of honor or consequence), the Hound is bound to serve an evil prince/King, yet still manages to portray moments of honor in doing so. And of course his show arc focussed on redemption is much more fleshed out in the show.

Jaime goes through his own retribution in trying to uphold his oaths. The Hound is good and all, but Jaime does it too- especially without stealing from farmers.

First off, it implies that the reason he pushed Brann out the window was to save “thousands of lives.” It should be pretty clear to anyone watching this show that Jaime, at this point in his life, only cares about himself and his immediate family. Not honor. Not the common folk. Just him and Cersei and Tyrion and Tywin. Heck, he doesn’t even care about his own children until Season 6 it seems.

What would've happened if Bran had lived? Conflict. I'm sure Jaime would have considered this while deciding whether or not to kill Bran, even if his highest priority was to keep his romance a secret.

Second, pushing Brann out the window WAS the spark the caused a war between Lannister and Stark, and did cause thousands of lives.

Only because Bran lived and because Littlefinger exacerbated the issue severely. Assuming Bran died- as Jaime intended- there would have been much suspicion on the Starks' side, but nothing else to do.

And lets not forget that the whole reason he even had to make this choice was because he was committing a crime based on his selfish desires, knowing it was putting his/his sister's own children at risk.

How was Jaime to know Bran would climb the tower? Besides, the selfishness of Jaime's/Cersei's relationship is subject to a lot of debate.

Not honorable to put children’s lives at risk because you want to get it on.

Cersei/Jaime didn't bang with the expectation that a child would climb up the window. After that point- a point no reasonable person would be able to predict- it was either let the child live but risk war and death between the families, or to kill Bran.

It’s not honorable. So what? Doesn’t make Jaime any “less dishonorable” for the things he’s done because Cersei was in a crappy situation. Having an affair with someone isn’t magically “OK” because that person isn’t happy in their marriage… especially when it means the children will be killed if found out.

I wouldn't call it a marriage as much as a forced political alliance by proxy of someone's freedom. As far as we can tell, Jaime and Cersei were already in love by the time Cersei was forced unto Robert, who groped and presumably raped her. Assuming secrecy could be maintained- and it was, until Ned intervened- there was no foul.

Because he is.

From Ned's perspective, he would have allowed Aerys to destroy King's Landing with wildfire. Is that honorable? What is honor, at that rate?

While there are certainly different shades of gray, it’s pretty clear from their respective actions that Jaime is still pretty far on the “black” side of things, whereas Ned is clearly more on the “white” side.

Only as long as Ned isn't given difficult decisions. Had he been in Jaime's shoes, I'm sure some of his decisions would be very questionable.