recursive lambda expressions
Informally speaking the two x's are indeed different variables (technically speaking, they are the same variables but bound by different abstractors).

As for your two inquiries regarding equivalences: in the strictest sense possible, they are not syntactically equivalent ($M \not\equiv N$). However, they are what is called equivalent modulo alpha ($M \equiv_\alpha N$). This basically means that one can be transformed into the other through $\alpha$-conversion (which leaves the behaviour of the term unchanged). The result is that the names of bound variables become arbitrary - as they should be.

Note that most equivalences in lambda calculus are stated modulo $\alpha$, and most lambda logicians seem to regard the formulation and application of the rule as a tedious formality. Once they have specified the $\alpha$-rule (if they even care to do so in the first place), they tend to adopt the following convention: 'from now on, when we say $M$ and $N$ are equivalent ($M \equiv N$), we just mean that they are equivalent modulo alpha ($M \equiv_\alpha N$), unless stated otherwise'. So in most contexts, you can simply say they are 'the same', but I think you should be aware of the details behind it.
Yes. If $y$ were to be substituted for some $z$ in that term, then all occurrences of $y$ will be replaced by $z$.


This is perhaps a useful free resource that covers some of the basics a little more rigorously: http://www.cse.chalmers.se/research/group/logic/TypesSS05/Extra/geuvers.pdf.