Hard starts and stalling get progressively worse

There are a few issues going on:
Difficult to start and will stall immediately after start until it has warmed up a few minutes; also, fan running constantly
-- Upon first start, vehicle quick to start but dies out almost immediately. Next time to start is more difficult, and gets progressively worse every time. After a few starts the only way to get it to start is by pumping the gas while cranking it.
-- After starting, engine will die almost immediately unless foot is held on the gas pedal. After a few minutes the engine is able to maintain idle on its own. Then the car is drive-able. (If driving the car before it is able to idle on its own it will stall when driving in low gear. Car is automatic trans.
-- When idling on its own, it's rough at first (it gets better as the car is warmed up)

Another issue (maybe related?) Is that the fan is running constantly. I was told that the temperature sensor needs to be replaced (is this the coolant temperature sensor?) Could this be related to the start/idle problems?

Check engine light doesn't come on.