What type of recursion is this?
What is a name for ?

There may be a name for that recursion, but then I would have
forgotten it. It does not really matter. Call it central recursion or
middle recursion if you will. And you can even define it simply :


  A recursive rule (or non-terminal) is said to be middle recursive iff it is recursive, but neither left recursive nor right recursive.


If I have to write a paper, and cannot find the accepted name for
something, and there may be none, then I make one up and give a proper
definition for it so that people can read my paper.

The point is that not everything needs to have a specific name. You
give names only to entities and concepts that play a specific role in
the kind of work you are doing. You example seems to be just a general
case of recursion, which happens to be neither left nor right.

Left recursion is important as it plays a significant role when you
study left-to-right recursive descent parsers. Hence it was given a
name.

Regarding your second question, I would agree. This may even happen in
the absence of recursion, with a grammar generating a finite language.
$\qquad  A \to 0 0 \mid 0 1$
But there are other grammars for that language. This grammar is not LL(1), but the language is.