It's my understanding that in most years, all the snow melts. The glaciers are ice remnants from snowier periods (in other words, the glaciers are not getting bigger any more). I know, splitting hairs.
This is true. While there is year to year randomness, and some years we get growth, in general our glaciers are receding. And some, disturbingly fast. Snæfellsjökull - the one visible from Reykjavík on a good day - probably only has a couple decades left.
A google image search for "Iceland highland roads" shows a bunch of roads from Iceland's highlands. Obviously. But almost all of them look very tame by "off-road" standards.
It depends. I don't know what you saw when you googled, as Google's returns vary from place to place. The main north-south roads that get most of the traffic are mostly "bad" in the sense of being very washboardy and potholed, and having some fords to cross. For example, here's some random shots from one of them, Sprengisandsleið:
It's the side roads that are killer. For example, I mentioned in another thread that I did Gæsavatnaleið the other year:
I would not do roads like that in a Model 3. But my feelings are that with the bit more elevation provided by air suspension and larger tires, Model 3 could handle the tamer roads. So long as the washboardiness doesn't rattle it to bits

(it rattled my Ford's radio in half).
Which is very good because the Model 3 is not an off-road vehicle. Around here we call those "all-weather roads" which means they have a firm base that can support the weight of a vehicle even when saturated.
Thankfully, mud is not generally a problem here.

The country is too young to have much erosion to clays or other really fine sediments. In some places we have peat bogs, but generally not in the highlands. There's very little life at all in the highlands, to the point that it's sometimes eerie. You might come across a single plant growing on its own, some little flower, and think, "How did it get here? How can pollinators find it?" Or you're in the middle of this moonscape, and then suddenly come into a patch of lush, luxurious green covering a couple hectares... and then you move on and it's the moon again. The place really has an effect on you
The highlands threats are rocks, water, and isolation. Lots of all three
I would be somewhat hesitant to drive through standing water deeper than about 5-6inches
Gotta do that even on the major roads, let alone the minor ones that lead to specific destinations. And remember, some of the crossings (actually, the majority of them) are flowing. Crossflow - and your forward wake - means even more height. But this is on a car with what should be 7,4" clearance (very high suspension setting + larger tires); I wouldn't give a second thought to 5-6 inches of water, just tickling the underside. My worries start to creep in in the 9-12" range, and they become significant in 15+" range.
There is also the potential issue of being stranded in a very remote location.
Oh, very much. But the risk is part of what makes the highlands fun

Now, of course, getting rescued out there can cost over $1k... and there's "inventive" ways you can lose your vehicle. For example, the water on Flæður (several-kilometer-wide braided river) is only a couple centimeters deep on average - sometimes more, sometimes less. But it's packed with sediment, and flowing atop a deep sediment bowl, so if you just sit still out there for prolongued periods of time....
Or are highland roads fairly well travelled? Is there cell service?
Depends on the road. The major north-south ones, yes. The side roads to get to specific locations can be very little traveled, depending on the road. Cell service... surprisingly, yes, usually! We're a very connected country. When our last highlands volcano went off, there was live web streaming of it
He barely entered the water when it died. It turns out the Saab air intake was plumbed right down to the bottom of the car
***facepalm***
You might ask Tesla what the issues are with driving through standing water.
Yeah, if there's not much info forthcoming by the time it would be of issue, I'll pick their brains as much as I can. The hard part will be getting an answer from someone who actually knows the right answer. There's also some of the people with lots of teardown experience and those who salvage Teslas, like Ingineerix. I imagine people like him would have a pretty good sense of how well the car would likely deal with different water levels.
Maybe you should see if dad will buy you a Model X instead?
Hehe.. not in my budget, and I don't have interest in such a big "guzzler"

Plus it's not any more "offroad" (except for a better natural clearance) - it's not built to take a punch. And is probably a lot more expensive to repair than a Model 3.