I can give partial answers to 1 and 2. I think the viewpoint you mention in 3 is taken in very many references, so I'm not sure which ones you're using.

Perhaps the main reason for considering only bounded intervals is that numerical analysts are interested in provably (pointwise) convergent schemes. At least for traditional methods (Runge-Kutta, linear multistep, and many others) the step-by-step nature of the approximation precludes the possibility of uniform convergence on unbounded intervals. That is because the local truncation errors from each step accumulate -- not in a catastrophic way, if the method is stable, but still in general they accumulate and the error grows without bound as the number of steps taken goes to infinity. So uniform pointwise convergence can only be proved for finite intervals.

So it's not that you magically run into trouble when you go past the end of the chosen interval. After all, the bounds of your interval don't enter into the numerical method, and it knows nothing about them. It's simply that your solution will be less accurate at the end of the interval than at the start, and this effect will continue as you integrate further.

There are some areas where very long time integrations are needed, and special techniques are used, for instance in the study of stability of the solar system.

Perhaps a helpful example is Exercise 7.1 from this set, although you should keep in mind that it involves a very poor choice of integrator for the specified problem, so the accumulation of error will not generally be quite so bad.

Of course, there's also the fact that if "one does not stop the iteration ... after a finite number of steps" then your computation will never terminate.

You didn't fully specify the problem, but based on the fact that you mentioned Euler's method, I have assumed that you're dealing with an initial value problem. If this question were about a boundary value problem, that would of course be a completely different matter.

Regarding 1 and 2:

Perhaps the main reason for considering only bounded intervals is that numerical analysts are interested in provably (pointwise) convergent schemes. At least for traditional methods (Runge-Kutta, linear multistep, and many others) the step-by-step nature of the approximation precludes the possibility of uniform convergence on unbounded intervals. That is because the local truncation errors from each step accumulate -- not in a catastrophic way, if the method is stable, but still in general they accumulate and the error grows without bound as the number of steps taken goes to infinity. So uniform pointwise convergence can only be proved for finite intervals.

So it's not that you magically run into trouble when you go past the end of the chosen interval. After all, the bounds of your interval don't enter into the numerical method, and it knows nothing about them. It's simply that your solution will be less accurate at the end of the interval than at the start, and this effect will continue as you integrate further.

There are some areas where very long time integrations are needed, and special techniques are used, for instance in the study of stability of the solar system.

Perhaps a helpful example is Exercise 7.1 from this set, although you should keep in mind that it involves an intentionally poor choice of integrator for the specified problem, so the accumulation of error will not generally be quite so bad.

Of course, there's also the fact that if "one does not stop the iteration ... after a finite number of steps" then your computation will never terminate.

You didn't fully specify the problem, but based on the fact that you mentioned Euler's method, I have assumed that you're dealing with an initial value problem. If this question were about a boundary value problem, that would of course be a completely different matter.

Regarding your 3rd question:

"given a discrete dynamical system, does there exist a smooth dynamical system such that the discrete one arises as its [discrete numerical] approximation)?"

This is often known as modified equation analysis and is a form of backward error analysis. For linear PDEs, it is described for instance in Section 10.9 of LeVeque's book. For Runge-Kutta methods applied to arbitrary initial value ODEs, it is much more difficult and is described most fully in Chapter IX of Hairer, Lubich, & Wanner. I think the last book is also a good example of one that takes the abstract viewpoint you are wishing for.