Here's my take.

It's how I interpret the flags here as a mod and how I use them elsewhere on the network as a user.

###In which cases should we flag "not an answer" answers?

The not an answer flag does not mean "this answer is not an answer to the question posed".
It means "the post isn't an answer at all" (see the "It should possibly be..." bit of the flag description, although the "deleted altogether" bit is painfully ambiguous)

I'd extend this to include answers that are answering something completely unrelated to the question at hand (eg, Q:"How do I backup my data?" A:"Mix together 300g flour, 100ml water, 250g prawns and a ripe banana"), but these are pretty rare to see anyway.

Not an answer does not include a post suggesting something the OP has already ruled out, such as in your 2nd and 3rd example flags. In that case it's just a "bad" answer (see answer to 3rd question).

It's not cut and dry, but essentially if the post makes an effort to answer the question (even if it does a bad job of it) then it is an answer and this particular flag doesn't apply.

###How should we flag eligible answers? What do we put as the close reason to be clear?

I assume by "eligible answers" you mean those like I've outlined above as being not an answer?
And I'm assuming "close reason" is a typo for "flag reason"?
If those assumptions are correct then:

Just flag them as not an answer. If you feel they're an odd case then flag them as other stating "not an answer; because x y z" or something similar.

###What do we do when the answer is ineligbile for flagging, but isn't a good answer?

First, I'm assuming there are no other problems with the post. Obviously if it's spam, offensive, appalling quality, etc you should flag it for whatever other reason might be suitable.

Assuming it can't be flagged for anything then the answer is valid within the confines of the system, it's not an exception and doesn't need moderator attention.

Add a comment explaining the problem (or upvote any comment that has already done so).
If it's particularly bad for some reason then downvote the post.

Here's my take.

It's how I interpret the flags here as a mod and how I use them elsewhere on the network as a user.

In which cases should we flag "not an answer" answers?

The not an answer flag does not mean "this answer is not an answer to the question posed".
It means "the post isn't an answer at all" (see the "It should possibly be..." bit of the flag description, although the "deleted altogether" bit is painfully ambiguous)

I'd extend this to include answers that are answering something completely unrelated to the question at hand (eg, Q:"How do I backup my data?" A:"Mix together 300g flour, 100ml water, 250g prawns and a ripe banana"), but these are pretty rare to see anyway.

Not an answer does not include a post suggesting something the OP has already ruled out, such as in your 2nd and 3rd example flags. In that case it's just a "bad" answer (see answer to 3rd question).

It's not cut and dry, but essentially if the post makes an effort to answer the question (even if it does a bad job of it) then it is an answer and this particular flag doesn't apply.

How should we flag eligible answers? What do we put as the close reason to be clear?

I assume by "eligible answers" you mean those like I've outlined above as being not an answer?
And I'm assuming "close reason" is a typo for "flag reason"?
If those assumptions are correct then:

Just flag them as not an answer. If you feel they're an odd case then flag them as other stating "not an answer; because x y z" or something similar.

What do we do when the answer is ineligbile for flagging, but isn't a good answer?

First, I'm assuming there are no other problems with the post. Obviously if it's spam, offensive, appalling quality, etc you should flag it for whatever other reason might be suitable.

Assuming it can't be flagged for anything then the answer is valid within the confines of the system, it's not an exception and doesn't need moderator attention.

Add a comment explaining the problem (or upvote any comment that has already done so).
If it's particularly bad for some reason then downvote the post.