When I saw your photo of the antenna mounted on the pallet, it almost warmed my cold, dead heart.
Merely "sitting on", not mounted :-) I just wanted to get a few extra cm of height to get closer to the feijoa hedge and the house roofline being the joint horizon to the south (the most critical direction in NZ/Aus).
Seems to be working ok. I'll attach the obstruction map from the app. It maps which parts of the sky it doesn't receive expected signals from, so it knows to switch to another bird before that happens. You can clearly see the row of bamboo off the SW (left side of this screenshot)
1) Can you run a small server or two?
I'm not aware of any language saying you're not allowed to (and I'm sure not going to go looking for it). As with most consumer ISPs you'd need to use some form of dynamic DNS. I haven't actually been keeping track of my IP numbers but I've just now set up a cron job to do that every 15 minutes. I'll report back after a few days.
The provided WIFI router doesn't have any interface for setting up pinholes, but you are not required to use their router -- it just makes for a more foolproof OOBE.
2) What happens when the service gets saturated with users?
They are restricting the number of users in each "cell", a hexagon roughly 20 km across. Right now they are only allowing people in fairly rural areas to order the service.
There are currently 1288 operational satellites in the initial "shell" (set of orbits with satellites 450 km up) with a further 241 on "standby". I think the launch late January brought the total number of satellites to just over 2000. They are now building a new "shell" of satellites with sat-to-sat laser links that will allow your downlink to be far from you, rather than the maybe 1000 km [1] it has to be at present so that the same satellite can see both you and the downlink ground station at the same time.
Starlink have permission to launch and operate 12,000 satellites, and are seeking permission to increase this to 30,000.
That will give a lot more capacity than now.
It's never going to be able to cope with thousands of people in a city or city suburb. But those places have fibre and/or good 4G/5G signal.
3) How mobile is it? Can you or are you allowed to move the receiver to another property?
You can move house. But you can only move to somewhere where a new user would be allowed to sign up.
Putting it on your RV yacht and moving around every day is not currently supported.
4) Can one, as a ordinary user, examine the connection quality?
You can of course use the normal tools such as ping. The app does goes you a little:
[1] don't confuse this with the often publicised figure that you need to be within 500 km (or some other figure such as 600 or 680 km) of a ground station. That's assuming that you have only one nearby ground station and expect 100% continuous coverage. That means that at times the best satellite will be 500 km the other side of the ground station, and therefor 1000 km from you. I believe that even if you are, say, 2000 km from the nearest ground station, there will from time to time (about every 5 minutes worst case) be a satellite somewhere more or less in the middle, and you can get internet via it for a few seconds or a minute or two. That might be useful in an emergency or to do a batched upload/download, but you're not going to get the kind of continuous internet people expect in that situation.