How much DPS can carrier pull?

Carrier hulls are not that expensive, ~650-700mil; even cheaper if you build it yourself inside a pos. Compare this to the price of t3 ships often used to solo c3 wh sites. Carriers should be seen less as capital dps platforms (that's what dreads are for), and more as cap logistics ships that can also do dps.

A cheap way to do sleeper sites in a carrier is to use t1 large drones. The sleepers will kill a LOT of drones, but t1 drones are dirt cheap and carriers can hold hundreds. Carriers are also insurance against pos bashing gangs. Triage carriers can bring a pos out of reinforcement real quick, so you have to worry less about roving gangs hitting your pos while you were sleeping.

Carriers do anything from 700dps upwards (assuming 14 fighters), much more if you fit drone control units to allow more fighters. I wouldn't recommend DCU's in general; fitting capital logisitcs modules is a far better idea. Carriers are primarily support ships and should never operate alone. Thanatos does more damage than other carriers because they have a fighter damage bonus; but a weaker tank.

You don't necessarily have to bring your carrier to a sleeper site to use its fighters. They can be assigned to any other ship, and will attack whatever the ship attacks, and follow in warp.