Okay, I'm glad to know that there isn't a fixed fee to the airport. I misinterpreted the "standard fee" phrasing in the airport's website (and yeah, given that it's a german airport website, I realize it's probably not written by native english speakers). I'm wary of train station taxis; on our first (and only) trip to Vienna a couple years back, we arrived in the evening at the central train station. We showed the taxi driver our hotel's name & address. After a while, we questioned him about how long it was taking; he then immediately said the hotel is coming up soon, and the dropped us off at the WRONG hotel! It was the same brand, so we didn't immediately recognize our mistake until trying to check in at the reception area. Anyway, I got a lot more wary of taxi drivers (especially those at central transit areas) in most cities after that experience.
In terms of the tip or lack thereof, if the driver doesn't bother to get out of the car to help with the luggage, I don't tip (& this definitely includes taxis in the US too). In this particular case, the driver didn't help with my luggage (though I only had carry-on piece which I can easily handle myself so it wasn't that big a deal).
Yes, I know that the train would have been a lot cheaper, but I was running short on time and every minute counted. I didn't know when the next airport train would arrive, and plus there is always a longer walk from the train station vs being dropped off in front of the airport. Actually, this made me curious, so I checked out the train schedule. This was on a Saturday afternoon so the trains don't run as frequently. The earliest I could have arrived at the terminal via the train would have been 4 min before the check-in cutoff which would almost certainly have been too late.