Under what conditions do auto-turrets stay in play?

Solution 1:

I just put a few of these questions to the test.

You cannot have multiple turrets of the same type. I dropped a Viper Probe Droid, picked up another one and dropped the second Viper Probe Droid as well, and the first droid blew up as soon as I dropped the second.

Your turret will stay alive as long as you are alive or until it gets killed - The turret, after you die, has a limited time it will stay alive before it just blows up. Respawning does not seem to affect this timer - I've respawned to see my turret still activated and working before randomly blowing up.

Solution 2:

Turrets placed by you remain there until the end of the match, or they are destroyed. I've placed several manned and auto turrets on a particular map, and have died several times, noting that they are still there afterward.

You can have as many turrets (of the same type) as you want. You only need to pick up the powerup that grants them, and place them in valid locations. Note that they are favored targets by enemies. (I've had them destroyed seconds after placing them, most players know how powerful they can be.)