Thank you, this framework helps. Definitely no to 1. Definitely yes to 2, with some corrections. Yes to some parts of 3.

Re 2. First, let me adopt bounded realism here, with physics (external reality or territory) + logic (human models of reality, or maps). Let me ignore the ultraviolet divergence of decompartmentalization (hence "bounded"), where Many Words, Tegmark IV and modal realism are considered "territory". To this end, let me put the UV cutoff on logic at the Popper's boundary: only experimentally falsifiable maps are worth considering. A map is "true" means that it is an accurate representation of the piece of territory it is intended to represent. I apologize in advance if I am inventing new terms for the standard philosophical concepts -- feel free to point me to the standard terminology.

Again, "accurate map", a.k.a. "true map" is a map that has been tested against the territory and found reliable enough to use as a guide for further travels, at least if one does not stray too far. Correspondingly, a piece of territory is said to "exist" if it is described by an accurate map.

On the other hand, your "invisible mathy things" live in the world of maps. Some of them use the same term "true", but in a different way: given a set of rules of how to form strings of symbols, true statements are well-formed finite strings. They also use the same term "exist", but also in a different way: given a set of rules, every well-formed string is said to "exist".

Now, I am not a mathematician, so this may not be entirely accurate, but the gist is that conflating "exist" as applied to the territory and "exist" as applied to maps is indeed a category error. When someone talks about existence of physical objects and you write out something containing the existential quantifier, you are talking about a different category: not reality, but a subset of maps related to mathematical logic.

I am not sure whether this answers your objection that

why I should believe in these invisible eternal number-like things that exist in their own unique number-like-thing-specific way. (And what it would mean to believe in them!)

but I hope it makes it clear why I find your replies unconvincing and generally not useful.