1) Get the Armored Plated Shotgun. It's THE best tanking ranged right now and it's crafted by engineers. There's absolutely no reason for you not to have one.What gear should I focus on updating first?
2) Also, I'm not a fan of your bracers. Yeah I know RNG sucks but try to get the epic bracers from Heroic AK or the trash drop bracers from 10-man Naxx.
3) The gloves and chest can be replaced by T7 once you run enough heroics (or 10-man Naxx), something you should look to do.
4) Your belt can also be replaced either by the badge epic or by running Heroic AN for the epic belt off the last boss.
5) You can see if you can get a Signet of the Accord off the AH, it's a BoE drop off 10-man Sarth. Alternatively, try and get the epic ring off the last boss of Heroic DTK.
As you can see, running Heroic AK, AN and DTK everyday (if you have time) will eventually lead to you getting your T7 badge gear. But get the gun ASAP.
See above.Specifically, where can I get nice bracers?
It really depends on your existing stats and your healers. If your healers are complaining they can't keep you up, then sta and avoidance stats are always good. If that's not an issue, then +exp/sta and +hit/sta gems are good for building threat.What do I socket for?
It's a problem you will face as you start to upgrade from the blue gear to epic gear. When I first started getting Naxx drops, I had to constantly regem for +def to make up for the +def loss. My target was always to make sure I was at 540.P.P.S. It seems like all my blues have more defense than the epics that are dropping in 10-mans. Is there a general "rule" to use when trying to decide when to drop defense for raw avoidance and/or stam?
One very useful item to pick up is Repelling Charge in Naxx 10. It packs a lot of +def to it can be used to make up for the shortfall in defence. Don't be afraid to enchant for +def as well (my cloak and chest still have it).