Turing machine - infinite tape - does that thing exist? Can we use a Turing machine with infinite tape as a basis to prove anything disregarding the fact that such a thing can never exist? 
Do we have the right to regard a machine (a construct) in the same way we regard a number or a set (an abstract)?

If yes could I similarly use a Turing angel with infinite scroll instead, would that be ok?

EDIT: since I cannot comment.
This question is somewhat different from infinity in mathematics because a set even infinite can have a finite representation. On the other hand there is no finite representation for an infinite tape  That is the algorithm that generates the output.

Computation requires resources, that means that it takes place in a time-space, not in an abstract timeless dimension.
While it could be ok to draw on demand more time for the computation pushing on to infinity, space has a cap on the amount of information it can hold, and before reaching that point the tape would revert to write only, that is implode into a black hole. So even if an algorithm is "finite" its output could be big enough to become "unreadable".
For example lets say we have a machine that counts all numbers at some Number H when trying to add more information on the tape the tape colapses forms a black hole and number H+1 becomes unreadable. 
Counting up to H+2 is a finite process but cannot be completed.
One could argue that space and time restrictions should not concern us but I m not convinced that is true when dealing with objects embedded in spacetime

I could accept a definition of a machine with sufficiently large tape. But I think that the size does matter. 
Can for example 2 Turing machines with different tape-size considered equivalent?
The tape is not infinite. It is unbounded. 
There is an important difference.
We don't need infinite memory.
We only need to be able to add more memory over time as needed.
This is actually pretty close to what people do in practice.
When a program needs more memory it asks the operating system and
receives more memory.
If we see that a computer is short on memory we increase its memory.
It is becoming even easier with cloud computing.
So the assumption is not that the machine has infinite memory.
The assumption is that we can increase the memory as needed.

If your question is about the limit where 
the needed memory reaches the limit of information 
that can be stored in the universe 
(assuming that it is finite and bounded by some fixed number)
then yes, we will not be able to add any more memory at that point.
In fact, if you assume that 
the amount of information is bounded by a fixed number
then the world is a finite state machine.
However that is not the usual case.

However if you try to design programs as finite state machines 
you are going to have a very difficult time
because the number of states is huge.
The number of states for a Turing machine with $m$ bits of memory 
would be something like $2^m$.
Turing machines (and other similar models)
simplify things.
This is a rather common thing in sciences.
It is important to know the limits of a model but
the fact that a model is not a completely accurate representation of the world
doesn't imply it is useless.