My reply to people just starting off is to give 4 a try. It is much easier to know how to use all 4 and choose to just use 3 than the other way around.

I play with all 4 fingers, but at times find myself playing lines with a 3 finger approach because it works better than 4. In the same way I have guidelines for fingering that I often "violate" just because an alternate fingering just works better. For me I like to have a baseline (not bass line to start with and let it flow from there. My baseline starts with a 4 finger approach.

Just my thoughts on the original questions:

1. From what I've seen on youtube videos, it seems that it's completely possible to play the stick using all 10 fingers. In practice the thumb seems to be used only for special effects (bass melody on outer string, harmonics, slapping, etc.) and only one thumb at a time. Does the instrument require thumb support? If it does, does it apply also to seated playing and/or an instrument stand?
For me, thumb support helps, at least in the left hand to stabilize the instrument. However, the Stick could be anchored against the shoulder instead in order to implement the left thumb, a la Boris Bazurov. A stand might be an option, though I don't have much experience using them.

2. On strings tuned in 5ths, would fingerings adapted from the cello work? For example for the two octave major scale (1 is THUMB):
1-2-4-5 / 1-2-4-5 / 1-2-3-5 / 1-2-3
I personally have difficulty using my thumb in such a manner, but I could see it being done. If you do end up getting a Stick, I might recommend looking at the 10 string grand model. It would probably give you a little extra space to work your thumbs in.

3. Would keyboard-like thumb-under technique work? Would it work for whole tones or just for semitones? For example the Minor scale on one string:
1-2 (thumb under) 1-2-4 (thumb under) 1-2-4
I don't think this would work. With a keyboard, when the thumb crosses under, the note is played by pressing down and you have a good half inch to create that velocity. Whereas on a Stick, the majority of the velocity must be generated prior to touching the string. You need to be able to tap, rather than just press.

4. On strings tuned in 5ths, how comfortable would it be to play a melody in octaves using two adjacent strings and fingers 1,4? How about fingers 1,5? How about adding a 3rd and a 7th to that chord with fingers 2,3?
Maybe on the outer strings it could work, but my thumb has difficulty hitting the inner strings. A lot of it has to do with the curve of the fingers and the angle of attack. The other 4 fingers have the ability to curve and reach over the other strings, and the tapping motion is up and down. However, the thumb doesn't really curve in that direction, and the tapping motion is more of a side rotation than an up and down motion. Also, keep in mind that the note only sounds while the string is pressed down. There's no sustain pedal like on a piano, so playing an octave melody with one hand might come out a bit choppy. Maybe with a reverb/delay pedal, you could smooth it out.

5. In your opinion, if I play with both hands consistently above the instrument in a piano like hand position, would a so called "uncrossed" string arrangement make more sense?
I'm not sure if it necessarily makes "more" sense, but I could see it working. A thought further, you could potentially string it such that the lowest notes are on the outer edge of the instrument, so that your thumbs have easier access to the bass and root notes.