The universe is computable.

Not self-evident. It may turn out that the fine structure constant is somehow related to Chaitin's constant, which is uncomputable. Or it could be a completely unrelated computable number.

All computations can be performed by Turing Machines.

If by "computation", you mean "something that can be done on a turing machine", then this is a tautology. If not, I'd say it's false. The halting problem cannot generally be solved on a Turing Machine.

The mind is made out of atoms.

Not self-evident.

The name of the empty string is epsilon.

That's a definition, not a self-evident truth.

1/ It will take me about a day, a packet of cigars and a machine full of coffee to write this program and start it running. That is what I am doing now. When I start it running, will I have done a bad thing? If someone were to stop me before the program started running, would that make any difference to anything important?

If you run it forever, maybe. If you only run it for the lifetime of the universe, it's not likely to do anything important.

2/ Can anything except what is computed by this program be said to exist in any sense?

Sure. That program only deals with things that are computable. If the fine structure constant is uncomputable, then it exists, but is not computed by that program.

3/ Does the ultimate Truth Table, which the finite tables TT(n) approach as the process continues, exist?

I doubt it exists in any physically real sense, but I have no way if knowing. Mathematically speaking, it can be said to exist. Some suggest that it existing mathematically makes it something real that can be experienced, but I disagree.

What does that question mean?

I assume you're either asking if it's physically real, or if it exists in some mathematical sense.

If the computational power of the universe is infinite, then it can contain not only itself but every other thing.

Not necessarily every other thing. A computer that can run an infinite number of steps in a finite time can be said to have infinite computational power, yet it cannot explicitly state every function from R to R. It can only have as many states as there are real numbers, but there are more such functions.

If the computation power of the universe is finite, where does that number come from?

You could work it out from how the universe works. If our universe was finite and non-expanding, we could calculate it by looking at the maximal entropy state and subtracting the current entropy. That is the highest number of unreversible computations that can be done in the universe. Unfortunately, our universe is a bit more complicated than that.

The program is very short. Any randomly chosen computation has a good chance of being it.

It's not astronomically low. It's still on par with winning the lottery. A sufficiently large program certainly has a good chance of including it, but we don't know how large our program is. Also, regardless of the size of the program, there's a good chance of it halting before it gets to the part where it calculates everything.

It would probably be very hard to construct an interesting universe which did not contain every possible universe and person.

No it wouldn't. A universe of finite size would work. For example, Conway's game of life on a torus.

6/ Does it make any sense to talk about 'not being part of this computation'?

Yes. Chaitin's constant is not part of this computation.