What type of recursion is this? I was trying to rule out whether this was LL(1) by checking for left-recursion in the following grammar:

$\qquad  A \to 0 A 1 \mid 0 1$

Which produces:

$\qquad 0 A 1 \Rightarrow 0 0 A 1 1 \Rightarrow \dots$

But I am unsure what type of recursion this is called, since it sorta stays in the middle.

If this is not left-recursive, would you agree that this is not LL(1) because non-terminal A has 0 appear twice in FIRST(A)?
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.