Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist? Does $1.0000000000\cdots 1$ (with  an infinite number of  $0$ in it) exist?

In the hyperreal number system which is an extension of the real number system you have infinite integers and the corresponding extended decimal expansions where it is meaningful to talk about digits at infinite rank (more precisely, rank defined by an infinite integer).  In this system your decimal makes sense.
Extended decimals were discussed in detail in an article by Lightstone:
Lightstone, A. H.
Infinitesimals. 
Amer. Math. Monthly  79  (1972), 242–251. 
