There's a lot going on in this question; let's break it down:
Is the assumption Con(PA) a philosophical one, and not a mathematical one?
I suppose it depends on what you mean by "the assumption Con(PA)":
If you're writing a mathematical proof, then either the foundations you're assuming prove Con(PA) or they don't, and it's a mathematical question which is the case. You'll be mathematically justified in assuming Con(PA) in the former case. In the latter case, there are contexts where "assuming Con(PA)" is mathematically justified -- for example, if you're trying to prove a statement of the form $Con(PA) \Rightarrow P$ for some $P$, then you're allowed to argue by assuming Con(PA) and proving $P$. But probably that's not what you mean.
If you're just asserting "PA is consistent" in a non-mathematical context, then your statement requires some philosophical unpacking to make precise, and moreover to determine to what extent it is justified. In this sense, it's a philosophical statement.
What principles lead us to believe that "Universes" is a safe assumption, whereas "¬Con(PA)" is not safe, regarding what we believe is "true" arithmetic?
There is an extensive philosophical literature discussing justifications for large cardinal axioms. You might start here. I'm not an expert, but I'm not aware of arguments which specifically argue that one should accept the arithmetic consequences of large cardinal axioms without arguing that one should accept the axioms outright. I would be interested to see such an argument. I once asked this question with related motivations.
I think the relevant thing to say is that I've never heard of similar sorts of justifications for $\neg Con(PA)$, and I don't expect I will.
(Next, repeat this question regarding the axiom of power set.)
Restrictions on powersets are studied in predicative mathematics. For some discussion of predicativism, one might start here or here. For some arguments for impredicativism, one might start here.
Is any theory that interprets PA "safe", as long as it is consistent with PA, and PA+Con(PA), and any such natural extension of these ideas?
By "safe", I take it that you continue to mean that the theory's arithmetic consequences are "true". There is a large body of literature discussing truth in mathematics. You seem to be particularly interested in the hierarchy obtained by passing from $T$ to $T + Con(T)$ iteratively. This hierarchy appears to be discussed from a philosophical perspective here. I think most logicians would probably agree that ascending this hierarchy adds relatively little consistency strength. And I'm not aware of serious attacks on the common-sense idea that if one believes $T$ is true, then one should also believe that $T + Con(T)$ is true.
Again, I'm not an expert, but it seems to me that most of the time when trying to justify stronger axioms from weaker assumptions, one appeals to some sort of reflection principle. This concept seems to be relevant to several of the questions you're asking.
You might also be interested in Reverse mathematics, where the "strength" of a mathematical theorem is calibrated by finding just how weak of a foundational theory can prove the theorem. One can interpret this as finding how "safe" a theorem is -- if you can prove it in a weaker theory, then it will continue to be true even if stronger theories are inconsistent, or simply "false".