Ooh, it's a biggy:
(I was going to also jump in on some of the logical parts, but I'll save that for later... rest assured there are many logical fallacies being thrown about)
If QM are so illogical to our logical understanding which we should not throw out in our day to day lives, how can we logically prove them to be true? What's more important? Which comes first? Logic and philosophy or mathematics and science? How can we understand quantum mechanics if they seem so illogical? If we can't truly understand quantum mechanics, how do we know logically that we are right?
Easy explanation. Quantum Mechanics is a mathematical theory. There are a ton of equations, and collectively they perfectly account for interations on the quantum level. And in keeping with all good theories, they could be used to predict lots of unobserved phenomena, which (despite seeming impossible) was later demonstrated to be true. The maths is perfect and completely correlates to observable physical phenomena. The only problem is that understanding the maths (in a real world sense) is a total mind****. The way the quantum world works is unintuitive to the way we see the world around us. But that's our limitation. You're not alone in thinking it sounds too crazy to be true. Schrödinger thought the real world interpretations of his maths was ridiculous, so proposed his '
cat' thought experiment to show this... unfortunatly for him everyone else thought this was a brilliant explanation so it got adopted as one of the main ways of explaining the interpretation of QM

Einstein also had big problems with the 'reality' of QM, he spent a long time trying to
disprove the Copenhagen Interpretation of QM.. however he was shown to be wrong.
How can I or anybody know Quantum Mechanics to be true if I and nobody cannot ultimately understand them logically in our minds?
It's unintuitive, that doesn't mean it can't be understood. Anyone (of reasonable intelligence) can understand it.
I personally believe that in order for these particles to come into existence, they need a cause.
That's fine, you can have whatever personal belief you want. Though that doesn't mean that it needs to be actually true.
Oh, and in my Chemistry we are getting into quantum mechanics a little bit. its quite interesting.
Cool, I didn't think it would come up in chemistry (but then I never really did it).
Asking somebody to believe Quantum Mechanics without a logical explanation, and saying "there is no logical explanation" is illogical.
Good thing I'm not doing that then

There is a logical explanation for QM, the maths is perfect, the way they related to real world events (the physics) is perfect. It's not illogical, it's just hard to understand intuitively.
There is no definite proof for the big-bang theory; please don't be dismissive.
That's true... in as far as we can say that nothing can be found to be perfectly true. Big-bang theory is just the best theory we have at the moment. And it is a very good one.
And particles simply 'popping in and out of existence' is not simple at all, and has as much scientific validation, as saying God created them.
Well that's true... you could say 'god did it' to absolutely everything. But if you're claiming to use reason then Occam's razor applies.. there is no reason to add supernatural premises to an event that can easily be explained through natural means.
The first and third are only observed in the surrounding area, most of the universe is presumed 'un-seen' Cosmic radiation proves nothing, but the observed patterns of the same (if not smaller) areas.
Oh, wait, you're claiming that the big bang theory isn't good? I'm afraid all the evidence backs it up, and no alternative can be thought of as being more likely. It would be unreasonable to suggest that it isn't the best understanding we have of what happened in the early days of the universe.
And about your claims that particles 'popping in out of existence' is a sound scientific principle?
Yes. Yes it is. Do you have a problem with it?
These particles MUST have a cause.
No, it's not the case that they must.
Unless you say that the particles are the uncaused cause of the universe.
There's no reason to jump to that - it was your argument that 'if an uncaused cause exists, then it must be the ultimate creator of the universe'.
But while you're at it, lets hypothesise that this is similar to how the universe was created. Instead of a single particle/anti-particle what if it was a huge number of particles/anti-paticles (they'd need to be balanced), though if it was a huge explosion the outward energy would need to be balanced, to fit the 'net energy of zero' conditions that virtual particles have. Well could the explosive outword energy be balanced... why yes it can, gravity acts as a negative energy... but if all the particles/anti-particles were balanced (this is called CP-Symmetry) then wouldn't they just cancel each other out straight away, leaving nothing? Yes... unless CP-Violations could occur in which particles could change from positive to negative (and vice versa) without breaking any physical laws - and hence leaving an unbalanced amount of particles (with which to comprise all the matter in the universe). Well, it just so happens that
CP-violation is perfectly possible, and has been demonstrated. There is a theory in cosmology that during the big bang, you only needed one 10 billionth of the matter to undergo CP-violation to leave enough matter to form all the stars and stuff in our universe. It's a pretty interesting idea that the creation of the universe could be very much like how virtual particles appear.
Nothing can come from nothing.
Well, there is a very interesting concept in M-Theory (The more recent development in String Theory), that there could be a greater 11-dimentional universe in which our universe is just one of many that is popping up (just like how virtual particles pop up in our universe). However it's very much limited to theory at the moment. The fact is we can never know for certain what could be outside our universe, if we can even say that there is an 'ouside'. There are predictions that could add weight to string/M theory, however we need particles accelerators more powerful than the LHC to do the tests.