You make a number of points here, and some of them conflict, I think. You can't both argue that we only experience a representation of the real world, and then on the other hand say that we have no proof that the real world exists. For a representation must be a representation of something - i.e. there must be something it refers to (represents), otherwise it is not a representation.
So, it seems you have two options - and I'm not quite sure which one you want to argue for. Either (1) Representationalism: there is a real world, but because we only experience a representation of it, we cannot know what it is like, or (2) Solipsism: there may be no such thing as a real world, or at the very least we cannot know that it exists.
If we end up with (2), then we have big problems, for we have no way of telling between what is true for me, and what is true for everyone. For instance, if I see a cat enter a room, and then it seems to disappear, how do I know the truth? What is true for me (my experience) is that I saw a cat and now it's gone. What I would normally do is look for an explanation that fit with general experience (mine and others'), but if my experience is only true for me, then I have no way of knowing what 'true' means. It might be that, in my universe (the only one that exists, as far as I know), cats simply disappear - or perhaps, sometimes they do and sometimes they don't. In other words, if I'm a solipsist, then I may as well be dreaming the whole time - weird random things may happen, because there is nothing external to my perceptions to guarantee that there is some underlying law or regularity. Basically, I cannot check my experiences against reality, because the only reality that exists is my own experiences! So, that's what's wrong with solipsism!
As for empirically testing, I just mean that I check my experiences against reality, or against other perceptions of it. So, the cat goes into the room, but then seems to disappear. However, from my other experiences, I have come to believe that cats don't just disappear, so there must be some other explanation (the cat is hiding, it left via the window, it sneaked past me without me noticing, and so on). As you point out, this doesn't give us ultimate proof, but it does at least suggest that there are some laws or rules which are independent of our experiences. We could of course argue that we are making up these rules as we go along, but the more complicated they become, and the more we find out things that we couldn't have created (or weren't aware of creating), such as maths and scientific principles, etc., then the more likely it seems that there is something which is independent of us.