Do this sentence mean a lot of feet and shoes not yet enter the room,
Probably, and one might assume so, but the literal meaning of those words is that many feet were starting to enter.
and will enter the room or
One might assume so, but the sentence does not exactly say that they will. It just says I saw them starting to enter.
are ready to enter the room?
Yes, probably, but again it says that many are starting to do so.
It's an odd sentence. It's particularly odd without knowing the context, how the feet are mentioned as well as the shoes, as if they might be two different things and the shoes might be entering independently of the feet - perhaps the writer could only see bare (?) feet and shoes? However the literal meaning, as written, is that I saw many feet and shoes that, at the moment I saw them, were just then starting to enter the room.
Could this to-infinitive replace to gerund like a sentence below?
I saw a whole lot of feet and shoes beginning entering the room.
Yes, but the double gerund is a very awkward way to say it. The infinitive "beginning to enter" is much better.