I embarrassed myself at a conference by asking an irrelevant question. Does this happen regularly?

If the question is "do most conferences have a few bad questions", the answer is certainly yes. People are likely to forget bad questions, more difficult to recover from a bad talk. I've been in quite a few conferences where someone (often a student without adequate guidance) would give a bad talk and everyone would be talking about it for days. I've also been to conferences where most of the talks were thoroughly forgettable and you would have to light the podium on fire to attract negative attention. I don't ever remember hearing people complain so vehemently about a bad question, only a bad talk.

The main "bad question" is one that is too basic for the audience ("what do you mean by calculus"?) or reveal a basic lack of understanding ("could you address heat dissipation by using logarithms?"). Questions that are too technical are rarely a problem unless it's way off-topic or takes up too much time.

I wouldn't worry about it. People ask all sorts of questions at conferences and no one in their right mind would conclude anything important from a single question you asked. Usually conferences are a blur as you try to take in too much information in too little time. There's just no time to waste on pettiness like remembering someone's embarrassing question - you're too busy remembering interesting research that actually matters to you - not to mention advertising your own work and generally networking. Normal people don't go to conferences to find someone to laugh at. Nobody will remember your "embarrassing" question in a few days.

Sometimes speakers will not like a question. Maybe the question sucks or is dumb, maybe they just have some personal hang up. It's silly to expect random conference attendees to know every speaker's personality and pet peeves so the latter case is unlikely to reflect badly on you. If your question sucks - oh well. Try to do better next time. If you feel it sucking as you're asking, make sure you at least keep it brief to avoid boring people.

Generally I try to phrase my questions in a very conservative, mild way so as to avoid any embarrassment from misunderstanding. Even if I feel I know exactly what I'm talking about and it is tempting to be very direct and pointed, I still ask as if I'm unsure of myself. I also qualify with something like "Sorry if I misunderstood something, I may have missed a few points in your talk" to give the person an out in case they don't feel like answering. If I wanted to publicly embarrass a speaker with my question, this is surely not a good strategy. But then I haven't felt the need to do that since... Oh, freshman year of college. As far as actually learning, this method of asking questions has served me well: People who have something interesting to say will ignore your qualifiers and address your point even if you did not demand them to. And if they have no interesting answer to give, you don't gain much from putting them on the spot.

I have probably attended close to 50 conferences or conference-like workshops (mostly Computer Science, but lately also a few in disciplines like automotive or education).

Does this happen in conferences?

Not really, at least not in the conferences that I generally attend. People do occasionally ask questions that indicate a certain unfamiliarity with the subject matter, but I have never seen the panelists give the asker a "highly judgmental, silent glare" or refusing to answer the question altogether. Under the assumption that your question was not monumentally stupid or offensive in some other way (and I have a hard time imagining this in your concrete scenario), I think the Occam's Razor explanation is that you are misreading the situation.

It's well possible that your suggestion was rather impractical, and hence the panel did not want to dwell on it for long (especially if time was already running out), but I cannot imagine that it was really nearly as bad as you make it out to be.

Or did I just make me and the company sponsoring my attendance really bad?

I would be surprised if the largest part of the conference remembered your question, let alone your affiliation, even in the evening of the same day. Conferences are long, many people ask many questions, and most of them are not particularly noteworthy. Again, unless your question was outrageous for some reason, there is nothing to worry about.

I looked online and saw people disapprove of questions that make it seem like the person asking is smart.

What people mean with that are "questions that aren't really a question". The ones were you are not looking for an answer, but are just trying to educate the speaker / audience. An honest question, even one that was maybe not directly relevant to the panel, does not fall into this category.