If I'm too hot while on couch and take sweats off in middle of night and have shorts that look like underwear can I get fired?

Working overnights are either "resting shifts" where a caregiver can sleep lightly or doze while her patient is asleep or "awake night shifts" which are, obviously, when the caregiver is expected to stay awake all night. When I worked hospice and had a resting night shift I would doze on top of of a bed or sofa, not get too comfortable, and just let my eyes close. I never fell into a deep sleep and I would pop up at the first noise. Your company should have told you what kind of night shift it was. And I agree that your directions come from your company, not your patient. Not with something like this. If your patient's family told your company one thing, your patient may tell you something else. Always confirm with your company. Many times patients don't want to put the caregiver out or they feel bad that the caregiver has to stay up all night but the company is usually hired by the family who relays to the company their expectations which oftentimes are different than the patient's expectations. Always check with the company.

As for taking off your pants, that probably wasn't the most professional thing to do even if you did have on shorts. I remember I job I had in the middle of summer and the couple didn't use air conditioning. They had air conditioning but they just didn't use it. I took a picture of the thermostat when it hit 85* and I was working and moving around and sweating my fanny off. Miserable! But old people get cold and they don't like to turn the air on. So I sweated out those kinds of jobs. I'd carry a paper towel in my pocket to wipe my face and took a cool shower when I got home. But it's probably not a good idea to strip down if it's hot. I used to think twice about taking off my shoes but after 9 hours I felt it was OK to do so if I was there for the night. Even then though I knew I was technically out of uniform.

Prolly out shopping for new shorts. ( sorry, couldn’t resist)

It doesn't look like Cybill is going to return with an update.

I would like to know how old you are? Because if young and new to the job, a little leeway can be made. If you have been doing this for a while, though, you should have known better. Lots of people work nights. So, they sleep during the day, Hard yes but not impossible. If you can't do nights, then don't accept the job.

I just don't know what to say.

If you can demonstrate to your agency that you understand why they have a problem with your dressing inappropriately and going to sleep when at work, maybe they will agree to keep you for a further probationary period.

Trying to be fair, maybe you haven't had the training you should have had in the basics of professionalism at work. But the most basic of all is to have some common sense. You're at work, obviously you can't swan about in shorts, and you certainly can't go to sleep - no, not even if your sweet little old lady client WHO YOU'RE THERE TO LOOK AFTER (that's the clue) tells you to.

I hope this works out for you.

Well she did say they were shorts that "look like underwear", not actual underwear, although I suppose that could still be pretty indecent🙄

Who is "she" who told you it was okay to go to sleep?

If it was a dementia patient, not the person in charge, so to speak.

Perhaps this just wasn't a good look - apparently sleeping on the couch in your underwear! Cwillie's idea about planning the clothing to be lighter would have to help. My knickers are more decent than most bather bottoms, but I still wouldn't wear them in public.

I think there is a little bit missing from your story but I'll answer your original question -
I used to clean houses, and often the people I cleaned for were little old men/women who kept their home much too warm for someone who was actively working. I knew that by the second visit so I dressed in appropriate layers so I could strip down to jeans and a tank top when I got overheated. It is just common sense.
Sleeping on the job is something that should have been discussed prior to your starting the job - in my opinion it is OK as long as you can wake easily if needed and you are still getting up to do the periodic checks expected of a night time caregiver. On the other hand some people feel that your shouldn't sleep on the job, and I personally would not.
Can you get fired for either offence? I would hope that your agency would be more lenient than that if it is the first time there has been a complaint, but the client may seek a different caregiver.