Ignorantsmith12 said:

TL;DR: To test my understanding of special relativity, I imagined a scenario in which a spaceship entered an impossible reference frame and then tried to calculate what the universe would look like to said ship. I chose an impossible reference frame to be absolutely sure what is and is not possible in special relativity. Please critique. By the way, I'm not actually in high school. I graduated a long time ago; I had to pick a prefix, and I am just that bad at math.

Assuming impossible things, in general, leads to contradictions. In fact, given inconsistent assumptions, it's fairly well known that one can prove anything one desires.

The classic example is "Given that 2+2 =5, show that I am the King of England".

I will append the details of the proof shortly, but I want to make my point as clearly as I can first. Making incorrect assumptions leads to nonsense. It's not a tool for learning. In fact, it's important to make every effort to avoid making incorrect assumptions, because, using perfectly valid logic, one can derive ANYTHING from inconsistent premises.

That said, here's a version of the well-known proof. (It's sometimes attributed to Bertrand Russel, but a quick online check suggested his version was slightly different, and the basic idea predates Russel).

Given that 2+2 = 5 , we can conclude that 4=5. Subtract 3 from both sides, then we have 1=2. The set of me and the King has two members,, but, we've shown that two is equal to one. Therefore the set of me and the King has one member. Thus, I am the King of England. QED. (Note that one can use the same argument to show that I am the Queen of England, the Pope, or anyone else).

So - to expand you understanding, you need to find another way, one that does not involve impossiblities/ inconsistent assumptions.

One example of how you might do this is the idea of "null coordinates". For a simple example, assign cooridnates (t,x) to a two dimensional space-time. (You can generalize the argument to a 4d spaccetime by assigning coordintes (t,x,y,z) if you prefer, but I'll do the simpler case.

The details about how to go about this are rather complex, at least the ways I know of doing it are. For instance, there is something called the Newman-Penrose formulation of general relativity. But that advanced treatment probably won't help you in your goal of understanding special relativity better. My personal opinion is that while there are lessons to be learned from this approach by borrowing some ideas from affine geometry, it's not useful to someone who is struggling to learn with special relativity.

If you are the curious sort, it might be interesting anyway to know that you can mark regular intervals along a null wordline (such as a worldine where u=constant or v=constant), but that while you can mark regular intervals along such a worldine, the numerical "length" of any such intervals will always be dependent on the observer. In the usual treatment of special relativity, we focus on things, called invariants, that are NOT dependent on the observer. If you are not familiar with this treatment, Taylor and Whereler's "Space-time Physics" is a good source for a treatment of special relativity that does focus on invariants.

But interestingly, one does not actually need to be able to assign numerical lengths to do geometry - there's a branch of geometry called affine geometry that treats this.. See for instance the wiki https://en.wikipedia.org/wiki/Affine_geometry

wiki said:

In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting the metric notions of distance and angle.

While I find this fascinating, I can't say that it will particularly help you if your main goal is to learn special relativity for the first time. Taylor and Wheeler's geometric treatment (in their book Space Time Physics) focussing on invariants is probably much more helpful if you're doing special relativity for the first time. I don't actually know your background, apologies if I've made some incorrect assumptions about your background and/or interests.