As a 68W on my first duty station (in Germany) am I more likely to be assigned to a combat unit or put in a sick hall position?

SFC Marc W. is correct. Nobody in RP can tell you the odds of you going anywhere that is more than a SWAG (well, it IS possible that the Chief of enlisted assignments for Europe just happens to be reading your post, but not likely).

You are a round peg. Depending on the round holes that are open when you arrive will determine where you go. This is influenced by who has a higher priority fill and frankly who has more influence with the person pushing the button ("Hey Phil, I REALLY need you to send me the next Combat Medic Specialist because the one we have now is leaving in a month and we want overlap").

THEORETICALLY, it will all be planned out and while they might not know exactly what the names of the round pegs are until they arrive, they will know that the next two round pegs go to Wiesbaden, then the next one to Ansbach followed by another to Wiesbaden and three to Stuttgart.

Again, THEORECTICALLY. In practice, you'll know when you get there.

It's not until you're a NCO or officer that you become a unique placement and start working with career managers (but even then you can be diverted, extended, etc... needs of the Army and all that).

I was assigned to LRMC from 2000-2003. I can tell you that if you get assigned there, you will not regret it. You would learn all kinds of medical knowledge that would benefit you if/when you get assigned to a Line Unit. Plus....LRMC is a unit that does NO FIELD TIME. The closest you'll get to the field while at LRMC is the qualification range. But, let's not count the chickens before the eggs hatch. You'll still need to get thru 68W AIT, first

A few things:
It's Sick Call, not sick hall. Every medic does Sick Call at/for their unit, that's what medics are for. Medics are being removed from hospitals across the world, the chances you'll end up in one are very slim

The person who assigns you doesn't care if you're single, it has no effect on your assignment. They have a hole in a unit to fill. They'll match your availability date to a unit that's losing a 68W1 at that time and that's where you'll go.

PCSing to Germany is not a deployment, it's an overseas tour. You'll want to avoid mixing those two terms up.

Your DS's don't have any information for you because that's not what they are there for. They don't handle assignments, or schools, or anything outside you in their BCT. If you ask your SDS exactly how and when you could apply for Airborne School after BCT you will realize that's not their area of expertise.

As for Airborne School, if you were denied at MEPS because you don't meet the medical requirements, you're probably not eligible to go. You didn't say what it was but it's usually colorblind for most people who get denied. There's a red/green light test you can take in the future if you want to try again. Sometimes Branch Managers will go to an AIT to ask for Airborne volunteers. They might come to yours, they probably will not. Even if they do, the doc will disqualify you again. If that doesn't happen, and you do get to Airborne School, you will give up going to Germany and you will be assigned to an Airborne unit. There are only two battalions in Germany where you'll find Airborne 68W1's. That's two platoons of medics, about a grand total of 30-40 68W1 who are all on a three year tour.

If you are sent to one of those two units in Germany as a non-Airborne person, the unit will send you eventually. If that doesn't happen, when it's time to reenlist you can reenlist for Airborne as well.