A gaussian functional integral is really just a regularized functional determinant which are well known to mathmeticians.
You’re being a bit unfair to Gaussian functional integrals.
There is a very pretty theory of Gaussian measures on infinite-dimensional vector spaces, which makes it clear that the NICE way to integrate on such vector spaces is not to use the nonexistent “Lebesgue” measure, but these Gaussian measures.
The technically sweet way to deal with this stuff is to use generalized measures instead of measures. Then any Hilbert space comes equipped with a god-given Gaussian generalized measure
exp(-<x,x> ) Dx
You can painlessly integrate any polynomial in finitely many variables with respect to this Gaussian measure. You can also integrate certain limits of such polynomials. This is secretly what we do all over the place in quantum field theory.
The term “generalized measure” may sound scary, but in fact one can rigorously explain all this stuff in a way that’s much easier than the usual explanation of, say, the Lebesgue integral.
In fact, I pretty much gave the whole story away in my last paragraph.
So it’s really not bad. The only problem is that there is nowhere you can read about this stuff without fighting through a thicket of overly erudite mathematics… because so far, only overly erudite mathematicians have attempted to make these functional integrals rigorous, and the Gaussian functional integrals are a piddly little special case of the monstrous machinery they’ve developed.
Anyway, once you learn about this stuff, you can see that regularized functional determinants are “really just” Gaussian functional integrals. I.e., the other way around from how you think about it.
Of course either point of view is acceptable, but personally I find it simpler and more satisfying to learn how to integrate and then see the regularized functional determinants pop out as answers to certain integrals - just like the physicists always said they would! - instead of saying “ugh, these integrals make no sense, so we’ll define them to be regularized functional determinants”.
Much as many people try to make it look like it, my impression has been is that interacting QFT is not just relativistic quantum mechanics and that the mathematical foundations are going to have to be something significantly different.
Time will tell. My personal hope is that bringing quantum gravity into the mix will actually make the math easier - at least, easier to make rigorous! Lots of people think this, for the obvious reason: spacetime discreteness of some sort may cure the ultraviolet divergences.
The trick is to get this to arise naturally as part of a quantum theory of spacetime, instead of inserting it as an awkward kludge.
QM was put on rigorous bounds years ago. There have been attempts at trying to axiomatize QFTs, but, AFAIK either one proves that the axioms are too restrictive and no interesting QFT satisfies them, or the axioms are broad enough to not be terribly useful.
ZAP!
Alas, the situation is nowhere so clear as this. The Haag-Kastler axioms very well could apply to the QFTs particle physicists actually study… but nobody knows if they do or not! With some some later additions, the Haag-Kastler axioms can be used to prove all sorts of wonderful things about QFT - just the sort of things one wants to be true. People like Buchholz have done a wonderful job of exploring their consequences. However, nobody has proved that any interacting theores in 4d satisfy these axioms.
Nobody has proved they don’t, either - except arguably for the phi^4 theory, which is still sort of controversial. So it’s completely unclear whether the axiomatic approach is on the road to triumph, or merely heading towards a mirage.
There is also a group of people who go about trying to construct some QFTs rigorously that satisfy some set of axioms or another, but they’ve really never managed to construct anything very physical. The last I heard, the most complicated thing was phi^4 in three dimensions. The standard reference on this is a >book by Glimm and Jaffe called something like Quantum Mechanics:A Functional Integral Approach. It’s very heavy on the analysis,
though.
Yup. It seems you’ve gotta prove lots of inequalities to get anywhere in this game… it’s hard work. The best result so far appears to be Balaban’s: existence of 4d pure Yang-Mills theory with a fixedultraviolet cutoff. By “fixed” I mean that for his arguments to work,
you gotta work in a box of some size and you *can’t take this size to infinity. I don’t know how big this box is, but probably it’s pretty damn small. 
Of course, nobody knows is this is a realproblem with Yang-Mills theory or merely a problem with his techniques.