Hi Eagle,
No such thing in the US. No such thing as a formal "slot", and terms like "slot improvement" mean nothing to your typical US ATC guy or gal.
You get your clearance over voice or PDC (via ACARS). You close up, call for your push back and start when you're ready (I can't think of a single place that requires you to call for a start clearance). In %99 of the cases, "Ramp Control" (Apron, on your side, I guess) is a function of the company running the gates at the terminal, NOT an ATC function.
Once you get to the border of the movement area (where the real ATC takes over), you call them, and you taxi down to the end of the runway, and when you are next in line, you go. First come, first served. Sometimes the Tower will jockey around the departure seuqnce a bit to even out the traffic on the departures, but even that is pretty rare.
Now, with that said, under certain conditions, some airports may have flow restrictions because of weather or traffic flow. In which case you will be issued a "Expect Departure Clearance Time" (or EDCT), but this is the exception, not the rule, and is situationally based. If you are blasting off from KMSP to KORD, and there aren't any TRWs or other foul weather, you're going to blast off when you get to the end of the runway.
All ATC in the US wants out of life is to get you out of their hair as quickly and safely as they can, which means getting you going in the right direction ASAP. Even if you have an EDCT, they are always trying to get you going early.
In the case of when the weather (%99 of the time caused by TRWs), most places have coded departure routes (CDRs), which will provide a near instant reroute that (a number of which will be pre-approved by your dispatcher), so there's no coordination with the company. Get the CDR around the weather, and you're gone.
ATC will also utilize low level departures if you're fat on fuel and the upper sectors are saturated (happens sometimes, again mostly TRW related, but I haven't seen that in a bunch of years). OR, if you're OK with staying below 10k for a while, you can even utilize a technqiue called "Tower enroute", where you can utilize a sequence of adjacent approach control airspace...you don't even talk to center.
My point is that US ATC has the goal of getting you going in the right direction as soon as possible, and there are any number of means that provide safe and legal separation. You may wind up at a slower airspeed, a different altitude, or get vectored for a bit, but you are off an running. Generally though, a different route, altitude or airspeed is yours for the asking.
Don't confuse the term "slots" when used in reference to certain high-density airports in the US, like KLGA. They only refer to a pair of operations (T/O & Landing), and not to a specific time.
Nu