This is science. The whole point of research is that we don't know the answer.

The way it work is someone comes up with an idea, and then it gets fleshed out into a hypothesis, and then people come up with ways to try and prove it. It doesn't always work -- sometimes it turns out the hypothesis was incorrect, and sometimes we just don't know how to prove it. Doesn't mean we don't try.

And frankly it's the same with the philosopher's stone. Just because we know now have the hindsight that it was completely bonkers doesn't mean they shouldn't have tried at the time. This is how science advances.

This article lays out some of the reasons why we think there might be one. Also, we've seen unification happen in the past. First, electricity was linked to magnetism by Oersted when he noticed that an electric current made a compass move (I believe he actually noticed this during an in-class demonstration). Until then, there was no reason to think that static electricity had anything to do with compass needles pointing north. Then, EM was united with the weak force at high energy back in the 70's. There are indications (more details in the article) that there are some strange coincidences in various values for strong, weak, and EM strengths/gauge groups that hint at the three forces becoming one at energies 10^15 times the mass of the proton.

An imperfect but perhaps useful analogy is the Higgs boson. Even though it had never been seen, a lot of stuff made a lot more sense if the Higgs existed, which is why we spent billions of dollars building the LHC, and indeed the Higgs was there.

They don't. It is how science works.

There are gaps in our theories where they don't meet up to suggest that these current models are incomplete or insufficient. There is no clear signpost that ever says "you've got it all". So new models and theories are developed and new experiments are made.

Some theories we definitely don't have the experiments to test (today). So they might remain unconfirmed for a while yet.

And even if a "next step" is ever found, there is no guarantee that these next steps don't point out even more things to discover.

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They don't, and GUT theorists have been struggling to find experimental confirmation for a long time.

That said, the standard model is incomplete in ways that GUTs might address, and that keeps the dream alive.

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They don't know. It's a belief based on what they take to be the structure of the Universe. It's debated topic within Philosophy of Physics and even within physics itself.

This belief comes from the fact that everything, so far, seems to be describable and connected mathematically. The problem is that in recent years we have seen that the physics that describe the nature of the very small seem to be different from that of the very big. This means that either i) we simply haven't found a way to connect both properly; ii) it isn't possible to connect both properly; iii) the fundamentals of our understanding of the universe are still somewhat incomplete.