Visualization of simple topological spaces

Question

I am not so good with visualizing some spaces and would appreciate if you help me.

1. Are my drawings of the topological spaces correct?
2. Why does $\mathbb{R^2} X S^1$ look like torus without boundary? (If it is correct). Could you describe, how to "fold and glue the $\mathbb{R^2}$ with the $S^1$ to obtain the torus? If this is non-compact, should I imagine "infinite" torus?
3. Do product topological spaces usually have product topology as "default" choice? Or what other intereseting topologies could they admit?

Thank you very much.

Remarks: In #4, I mean the closure of $\mathbb{R} \times [0;1]$ in $\mathbb{R^2}$.