It sounds like playing this way isn't "wrong" really, just uncommon?
I wouldn't even consider it especially uncommon, particularly for (although this seems somewhat backwards at first) taller individuals. As someone who is 6' 3" myself, I am absolutely more comfortable sitting with a lower stool, and I've actually found it to help my technique (particularly right arm) because with the endpin extended just a couple inches, I can set the bass to lean towards me more rather than being more vertical as it usually would be when sitting higher up, and this allows for me to more easily relax the weight of my right arm into either the bow or my plucking fingers if I'm playing jazz.
So part of the problem is just carrying this stuff around the building, and that bass isn't exactly light. Maybe it's a matter of having to walk so much while carrying this stuff that is making me uncomfortable while playing.
Being overly tense in general can be harmful to your playing, and you may not even notice it at times. I know there have been times when I had to carry my bass long distances repeatedly for a few days or weeks, and the initial soreness in the shoulder/arm/back from lugging it around definitely made a difference when I played. One of my saving graces thus far has been getting a wheel for the bass--perhaps consider getting a wheel like this one ( [Invalid or Expired Link Removed] ) to make getting around the building a little less of a chore. Something like that can really make your life easier.
I considered just buying my own stool and leaving it in my instructor's office, which I assume he would not mind but I'll ask. If not using a stool I can just use the desk chair or the piano bench in his office, either of those should be the right height.
This isn't a bad idea--but if you do it, I would suggest an adjustable one. As a student myself, I've adjusted my stool height many times over the years and think I've found something that I'll likely stay with for quite some time. However, if I hadn't had an adjustable stool, that would've required trying different stools at different heights, borrowing stools, buying new stools, etc. Would've been a massive hassle and made the exploration a lot more difficult, and seeing as you're still searching for a comfortable position, you may be exploring for some time before you find what you're really happy with.
One comment about desk chairs/piano benches--in theory that's a plausible option for someone who sits low, like me, but there is such a thing as too low with the bass. Being only a couple inches shorter than you myself, the bass itself is still too tall for me to sit THAT low with it, even with the endpin all the way in. My left arm would be pretty far above my head, right arm awkwardly bent to reach a good point of contact for the bow, etc. And if you try to lean the bass towards you more to mitigate those issues, you'll end up with the bass resting on the wood seam where the bottom meets the back, rather than the endpin. I don't know if that poses any risk to the instrument itself, but even if it doesn't, the wood is slippery and doesn't have any grip like the rubber tip or sharp point on your endpin, thus it'll end up slipping all over the place. As someone who much prefers to sit when playing and has ended up with no stool once or twice, I've attempted the chair thing and quickly decided standing would be more productive even though I've never been as comfortable standing and playing. Certainly, give it a shot, but you'll most likely want a stool that's a good 6-12 inches taller than most chairs. (For reference, the average chair is about 18 inches tall--the average barstool that many people talk about using would be anywhere from 24-30 inches).
I don't see myself keeping such a large object as a full sized bass in my little house, and still be able to play it with any regularity. I'd want something I'd be able to take with me to family gatherings and the like. I figure that if I buy this instrument while I'm taking lessons then I can learn on the instrument I'll be playing in the future, not the one I'd rent from the school and never see again after the semester.
If this is really the truth for you, then by all means go for getting an EUB. Most EUB's come with stands so that standing it very possible, by the way, so the thought that it would force you to sit isn't entirely likely. But if you think that's the instrument you want to play in the long run, get one and give it a shot. Ultimately, music should be fun, and if fun is the only thing you want to play bass for than do it in the way that makes you happiest, and induces the least stress. Hopefully your teacher would be understanding of the fact that you're doing this for your own enjoyment and not with the intention to try and win a symphony job, in which case doing a lot of your playing on a "real" upright would be fairly important.
You could also try a a variant that I have used. Sit on a "bar" stool with your left leg on a rung of the stool and your right foot on the floor. This position creates a natural place for the bass to rest- that is against your inner left leg. If you take your left hand off of the bass, it will still stay nice and balanced. And you'll still have the correct angle to play. Just angle so that when you are bowing on the E string, your right leg is not in the way. Plus these stools are relatively light in weight. The only downside to this- which isn't an issue for you at this point- is that it will inhibit the bass from vibrating optimal with your leg muffling the back. Especially if it fully carved. Good luck!
This is quite common--however, if foot pain is something the OP is already experiencing, this could be pretty uncomfortable for the left foot, especially if he ends up sitting on a lower stool. I found that the pressure of the rung on my left foot for long periods of time wasn't particularly comfortable for me, though obviously many other players do not suffer from this as evidenced by the number of people who play in this position. I suspect the issue has to do with my height--because my legs have to bend more in order to get my foot up to a rung of a certain height compared to someone with shorter legs, this exerts more pressure on the knee and also on the foot to maintain the more sharply angled position of the leg. Perhaps that's not entirely accurate--I'm neither a physician nor an engineer so my logic may be flawed there but it makes sense to me anyways. This is a big part of why both feet on the floor is my default sitting position now, as well as that it feels much more natural to have both of my feet aligned at a similar height, thus not throwing off my hips and lower back.
The thread David Potts suggested is really great--check it out. There are some very interesting approaches, a lot of similarities and key differences and just a lot of great food for thought when it comes to how you physically approach the bass.