linear algebra - Intuitive explanation of left- and right-inverse

I am reading about right-inverse and left-inverse matrices. According to theory if a matrix $A_{m\times n}(\mathbb{R})$ is full row rank, then it has a right-inverse. That is, $AC=I_{m}$. Similarly, if $A$ is full collumn rank, then it has a left-inverse. That is, $BA=I_{n}$. I have the following questions:

1. 
   

   Taking $AC=I_{m}\iff A^TAC=A^T \iff C=(A^TA)^{-1}A^T$ but this satisfies $CA=I$, contradiction. Similarly, taking $BA=I_{n}\iff BAA^T=A^T \iff B=A^T(AA^T)^{-1}$ but this satisfies $AB=I$, contradiction. How is that possible?

   
   


2. 
   

   Moreover, and most importantly what is the intuitive explanation of the left and right inverse? Is there any connection with the rows or collumns or any of the four foundamental subspaces of $A$?

   
   

Thank you very match!