Tactics
Run and gun is about one thing - traversing the most often-traveled areas of a map quickly (hot zones), to encounter and ideally dispatch as many targets as possible. If this is not your style (it is most people's), you best course of action is to play directly against this behavior - effectively, play anti-run and gun.
The first key element to anti-R&G that you'll have to worry about (and really this is an advantage all the time, but with R&Gers speed it's even more relevant) - you don't want them to know where you are. If an R&Ger's dot/indicator shows up on a minimap, it's probably no big deal to them - they're already dead, or running away from that spot. You, however - you're a prime target for all the R&Gers in the area, instantly. Another key point here is that camping is probably a bad idea - you might get one kill, but there's a good chance that player is going to come looking for you.
The second key element is knowing where they are (or will be), and using that to your advantage. Getting to know the maps very well is incredibly important - specifically, the spots that are less often traveled, have good cover, where you can't be easily seen by someone moving quickly, or that allow good line of sight to major walkways (without being in a major walkway themselves) are the best. You don't want to stay in these spots long, however - after you get a kill or two, head in a safe direction under cover to another good spot.
Equipment
Weapon Proficiency: Silencer
Obviously you'll need to select a weapon that can get a silencer. The key here, obviously, is that since you won't be able to move quickly away from a location you fired at, you don't want the shot(s) to show up on the minimap.
Perk: Assassin
A similar circumstance - UAVs are an R&Gers best friend, any way you can keep off their minimap is a good way to play. Consider Blind Eye as well - R&Gers often get some nasty killstreak rewards if the rest of your team isn't prepared.
Lethal Equipment: Claymore, Bouncing Betties
Once you know the hot zones, this is where you get to have some fun - if there's a good bet an R&Ger is going to come around a particular corner, make them regret it. You can also use these defensively - holing up to snipe with one of these at the entrance is always fun (though many of the good sniping areas have multiple entrances in MW3).
Survival mode never ends. I have found the best strategy thus far is pick an area you can defend easily next to 1-2 buy points, preferably ammo and grenades crates.
It needs to be an area with overhead cover to avoid choppers, ideally should only have 2 entrances at most. I have yet to find a good spot on maps that have 1 entry point.
Get the sentry granade launcher, have your partner do the same, Get Riot Squad reinforcements,
The idea is you can stay behind the sentry guns and riot troops and kill everything, You have Sentries guarding 1 entrance and the riot troops at another.
You will need explosives later on to take out juggernauts, normal enemies will start to take a whole clip to kill around lvl 30 if you dont got for headshots.
C4 and Claymores are helpful but can get a little expensive to buy every round.
Flashbangs are invaluable for taking the juggernauts out. By mid lvl 30's you facing 4 juggernauts, and exploding dogs with armor on most maps. It can get crazy. I have not found any strategy of displacing to be effective, the enemy troops will claymore most doorways and choke points.
Perks - Longer Sprint was the most helpful I found.
Best Answer
As far as I can make out they last indefinitly until destroyed. I have had sentry guns last up to 30 rounds before they were taken out.
The enemies WILL target your sentries and this also changes their pathing and stradegy. Ive had enemies ignore myself and\or my partner and beeline for my guns attempting to knife them. (such an attack IS an instant kill on your guns and will likely be the way the enemey takes them down)
Dogs, it should be interesting to note, dont seem to properly register your sentries, and will often get 'stuck' on them, running in place until they are shot dead, or you move and the AI for the dog changes its course to run at you.
I barely use Grenade Launcher Sentries but the few times I have used them they seemed somewhat decent, if albiet extremly slow to shoot.
Normal Sentry Guns however, are golden. Brian has his opinion and he is entitled to that, but I have found that S.Guns, WITH PROPER PLACEMENT AND SUPPORT, are one of the strongest support equipments available and will more than pay for themselves in short order.
Note: IT IS EXTREMELY IMPORTANT THAT YOU EXPERIMENT WITH WHERE YOU PLACE THEM. Improper sentry placement\support will make them nothing more than an expensive waste of time. This is doubly so for juggernauts who tend to go through Sentries like a hot knife through butter. You might want to lead the juggs AWAY from your guns if possible, but the choice is yours of course.
Edit-Afterthought: I should specify that the guns work best when you support them. Left by themselves the guns will inevitably get overrun and destroyed, but when you work together with them is when they truly shine.