Well, your providers are correct.

But...

Lamotrigine can cause nausea by itself, though it tends to fade after maybe 6-8 weeks. It isn't exactly the most common side effect, but it happens.

And, switching meds in that general category (stuff that changes brain activities) can cause nausea, even if the drugs themselves didn't. Buproprion is one where stopping it comes with some degree of side effects for some degree of people (reports vary a good bit, but there's too many to dismiss).

Estradiol shouldn't cause nausea at all, but humans have weird reactions to hormones ringtones sometimes. I know a kid that has extreme menstrual cramping, to the degree of it causing incontinence when they happen. She's tried every hormone med out there, and estradiol was one of them, and she did experience nausea during that time. But how much was the hormone, and how much was the overall situation?

On paper, there aren't any medication interactions between any of those drugs. This doesn't mean that they can't end up having indirect interactions. There's not a ton of research afaik into the real underpinnings of things like depression and testosterone levels, but there is some interplay that happens. So, if you take a med that changes your brain chemistry, and that causes your T levels to change, what is the actual cause? I'm not sure there's a definitive answer to that, just that it's something that people report happening often enough to be interesting. Anecdotal reports aren't great for medical research, but it is enough to make it worth researching eventually.

Which is a long way of saying that they can be right, but you could still be experiencing unusual things that are just so unusual that there's no way to be sure what the actual cause is.

As far as your last question, a lot. I'm cis, but because I used to be a nurse's assistant, I end up getting calls to explain stuff from lab reports and medical records by pretty much everyone I know that isn't in the medical fields themselves. So, my trans friends tend to be a bit more open about the medical stuff than they might be with someone else. Now, limited numbers, but I can't think of any trans people that didn't deal with some degree of depression and/or anxiety. Not the ones I knew/know well enough to talk about that kind of stuff anyway. Transition does seem to reduce severity as things progress, but it's a damn near universal thing.

Here's what I would tell someone that came to me irl. Track your intake, sleep, and medication time for a week or so. Meals, drinks, and any OTC meds. Track your nausea levels throughout the day as well. See if there's a pattern. Sometimes, that's the source of an idiopathic symptom that shouldn't be medication related; when the meds are taken and how. If you see a pattern after tracking, then you've got a solution. And never rely on memory, track it. It's too easy to forget how much tea you had when you took your meds. Does taking only a sip change things compared to a full glass? You don't know for sure until you track it.

While you're doing that, request an anti-nause med. They're not addictive, they're not going to interact, so most doctors are fine prescribing them. It's a short term solution, and you'll still want to track things before you start taking them, but it can take a while to get the prescription and get it filled, so might as well ask at the beginning.

Once you track for a while, try changing something minor. Like, what you drink when you take the pills, what and/or how much you eat before or after taking them. Or time of day, though that's the last thing to change because it's more disruptive, and never change anything that's instructed by your providers. If they say to take something with food, never change that. If they say before bed, never change it. Well, never change it without consulting them, because they have a reason for that instruction.

It can take a month or so, but if you do the tracking and changes, and don't find anything that helps, you'll have the anti-nausea med as an option hopefully.

There are nausea reducing things you can use that aren't prescribed, but it's usually better to try and pin down the problem instead. Ginger really does help. Pedialyte or emetrol kinda helps, as does cola syrup. Peppermint can work, though I find it less effective than ginger. Just don't get into the kind of stuff like Dramamine without talking to your doctors.

Cannabinoids would fall under the consult first category. It isn't that there's any known interactions there, it's more that doctors vary on how they handle the matter, and there's good reasons to be very careful introducing them into your system when you're already dealing with an atypical situation that involves a medication that interacts with your brain chemistry. Cannabinoids are really low risk, but not zero, so you want your care team on board all the way. They're also kinda overkill tbh. The level of nausea you describe is really more suited to something like ginger because you always default to the least broad option. Why take something that works across your entire system when there's options that work directly in the stomach?

I dunno, I feel like that's a lot of text on screen already, so I won't go into anything else, but I'm fine with explaining any of that more, or looking stuff up for you, or whatever. I hate to think of someone dealing with that kind of nagging symptom when it is something that can be fixed with a little help, so whatever I can help with, feel free to ask.