Questions about time and space (from beginner)

A)

Is it right that what we see/feel in every moment is projection of
some 4D reality into our 3D world noticed by our senses.

It's more similar to a slice, hyperplane or hypermanifold, less to a projection.

If my eyes were able to "somehow" see in 4D and I pointed my eyes to a
person, would I see that person in all stages of his/her life from
birth to death as a single "entity"?

In a classical world, yes.

B)

Can I put infinite number of 3D spaces into a single 4D space?

Yes, a possible mathematical description of such a space-time is a 3+1-Minkowski space-time.

If we considered that 5th dimension exists and that we could put
infinite number of 4D spaces into 5D space, then we would end up with
infinite number of parallel realities, in which Caesar did not crossed
Rubicon, or did not even exist. Is it right or complete nonsense? Can
be somehow proved that this is right/nonsense?

That's close to the many-world interpretation of quantum theory.
But it's thought of as an infinite-dimensional Hilbert space, as one of the simpler ways of thinking.

Thinking of a 5D space-time as a collection of all possible 4D space-times is less suitable, since it would be a mere set of 4D space-times without meaningful structure. Take Ceasar crossing Rubikon a day earlier or day later or a meter to the left or a meter to the right with or without a leaf falling from a tree, or a huge number of other versions. Which would be the correct order to pile up these versions to a 5D world? The infinite-dimensional Hilbert space model solves this ambiguity, and provides a meaningful metric for the set of all possible worlds.

The 10-dimensional models of some string theories are something else: The 6 extra-dimensions are thought to be curled up, and tiny.

We have no access to parallel worlds, so it cannot be proven by observational evidence. It's a mathematical model. It's possible to prove theorems within the mathematical theory, but that's usually based on axioms, or assumptions. Mathematical theories may, but don't need to match with observation, depending on how the axioms match to physical "reality".