Okay here we go folks this is my big post on Season 3 and why I did not find it morally ambiguous (note not the same thing as saying I did not have fun playing it) and your results may vary in which case I would love to hear about it, since I'm always interested in hearing how different groups dealt with the problems /runs my group has.
Anyway, here we go on a run by run basis
Everyone is your friend: This one fails at being morally ambiguous because NOTHING in this run seems to resemble reality as I knew it. You get hired by a guy who is powered by love because the woman he loves has run away so you have to get her back, and he never makes it clear why she did it. Ghostwalker kicks you out of Denver despite the fact that the last time my group saw Big G we had decided to drive our Tour Bus come Armored Personnel Carrier through the streets of Denver in the process blowing away the elite shock troops of several of the groups that want to undermine his rule so that we could return his magical macguffin to him and also having said armored personnel carrier get its ARO properly registered as a ZDF prior to said drive so it's not like its our fault several gangs decided to launch major attacks on a government vehicle. In short, Ghostwalker should have no reason at all to be mad at us. Which brings us back to my thoughts on this run which are in short because it needs to serve as a bridge between season's two and three everything ends up feeling awkward and out of character so who is surprised if the runners act that way (by kidnapping someone) also.
Ready Set Gogh. This run fails at being morally ambiguous because your actions lack repercussions. Yes you are depriving the public of the chance to view a certain piece of artwork by stealing it and giving it to someone else, but considering all the shit most ordinary people in shadowrun have to deal with, this is not exactly stepping on the necks of people who are trying to form a union. Also you end up stealing the piece of artwork back and a few extras along with it because some other rich prick wants to dick over the original rich prick who hired you. So in actuality the museum up stole from will probably end up with more artwork than it started with!
Block War: This one fails at being morally ambiguous because the runners are so clearly in the right. First you do whacky frat hijinks which is just fun and not morally relevant one way or another. Then a little girl's father (your newest fixer) who was her only parent at the moment gets murdered. If the runners have any morals at all they will dedicate themselves to tracking down her murder. My team did and we then managed to hack basically everything with a camera in the area where they dumped his body. This gave us a couple of clues (a license plate and a face) and thanks to a combination of matrix and magic searching we tracked down the responsible party. Then revenge happened. Then we stepped back onto the rails and did what was left of the adventure easily because the guy who was supposed to try and backstab/murder us halfway through what was left of the run and his goons had already been killed for murdering our fixer. If a runner team doesn’t care about orphaned little girls then they're incapable of moral ambiguity since they're clearly evil bastards.
Burning Bridges: This mission is not morally ambiguous because it effectively takes all the options out of the players hands. If you're going to do the run then you're going to be terrorists, there's no way to white hat it the best you can do is try to minimize casualities. Also it doesn't matter if you choose to betray Mr. Johnson or not because even if you don't Aztechnology will still see to it that he gets blamed. Moral ambiguity implies different choices with different outcomes in my book, here you are either super mega evil (blow up bridge during the day) or just really evil (blow up bridge at night).
Monkey Wrench: This mission is not morally ambiguous because you're not really doing anything wrong. The fake break in thing is perfectly legal and it effectively makes you Horizon security contractors for the sake of the run. You use non lethal ammo so no body gets hurt worse than a light tazing or gel round splatter. Then of course the floor explodes. At this point there is nothing morally wrong with just tazing down anything from the Ares facility that enters the Horizon one, since they are trespassing without horizon's approval to try and trespass to test security like you are. Then you can leave and when Ares checks your vehicle they won't find anything of theirs in it so they'll let you go. So long as you actually do the job you're hired to do then what have you done that is in any way objectionable?
Inn -N-Out: this mission fails to be morally ambiguous because Aztechnology is the biggest dick in town. You start out having to defend a girl's restaurant chain from a hostile take over by them and you are clearly doing the right thing. Then you need to see to it so that her mentally not quite all there brother can avoid serious jail time by putting the blame on Aztechnology. This is only fair because Aztechnology framed him first regardless of who was truly guilty (and probably would have blown up the bridge themselves if they didn't see someone else about to do it). While you're in the prison you also end up most likely saving his life from another shadow running team that is trying to assassinate him, another morally laudable act.
Jackknifed: Fails to be morally ambiguous because you're clearly in the right the whole way through. You start out fighting a gun battle in complete and total self defense. You move onto being bodyguards for someone whose boss may be trying to kill them. You end up play Robbin Hood stealing from the rich (and the Aztechnology associated rich at that) to give food to people who are starving, not a moral choice I'm going to be up all night pondering.
Knight at the Opera: Fails to be morally ambiguous because you're effectively doing two jobs; one of them is body guarding, the other is sneaking a guy out of an opera so that he can go watch a football game without anyone realizing it. Willing extractions don't ping my moral barometer, especially when part of the job is not to hurt the security detail you are extracting him from and then returning him to.
Firestorm: This mission fails to be morally ambiguous because all the non PC major players in it act like dicks. They don't trust the shadowrunners they hire to do the job they were hired for and insist on sending their own trigger happy agents onto the scene. When the people who hire you betray your trust like that, there's nothing morally ambiguous about betraying them right back.
Now for Something Completely Different: Fails to be morally ambiguous because you don't break any laws in this one either honestly. You get hired as body guards. You body guard. You rough up some artists who effectively kidnap all the people at the art show. You go on mystical vision quest. You convince/bully spirit into stopping his bullshit before somebody/even more people get hurt, all in a good day's heroing.
Spin Control: This one fails to be morally ambiguous because bug spirits are the shadowrun equivalent of the Borg, except that you can free a Borg from the collective, I've never heard of anyone having a bug spirit cast out of their body successfully after being possessed. So I don't cry for the bug spirits that are being dissected and experimented on by Ares anymore than I do for the Borg that get torn apart/blown up by Species 8472.
Food Poisoning: Fails to be morally ambiguous because once again Aztechnology is the biggest dick in town. Someone (Aztechnology) released a bio weapon to the public as food. They are clearly evil. You are trying to track down who it was and bring them to justice, you are clearly doing a good thing. Only possibly morally ambiguous part of this was Breeze, the crippled shadowrunning technomancer who works for Aztechnology luckily we resolved the issue of if it was okay to tip over a cripple's wheelchair if they're a technomancer by having our parties technomancer (who is also paralysis from the waist down) do the interrogation. Never before have his lack of functioning legs proved such a boon to the party!
Just to properly establish the goalposts, one of the best examples of moral ambiguity I've had while playing Shadowrun was during the 2012 Scramble at Origins. Our mission was to find a runner get him to a safe house where he will be met by a cop to make sure he stays safe. Halfway through the mission we learn that the cop we're going to be handing him over to is a dirty cop, this makes me wonder, are we turning this well respected runner over to somebody who is going to see that he is killed or is he the kind of dirty cop who likes shadowrunners? Are we helping save this runner's life by getting him to a safehouse, or are we getting manipulated behind the scenes to make it easier for him to be killed? When you're not sure what the right thing to do is, that's when you're waist deep in moral ambiguity.
And there you have my thoughts.
Also on the subject of inter party bickering, just remember this important lesson, Stick & Shock and Stunbolt means never having to say you're sorry or write up a new character sheet.