1. If this neighbor decided to remove the shared property line section
of fence that is falling apart, and replace it with a new length,
are you required to chip in with the cost?

Under most circumstances, yes.*


2. If the neighbor puts up this new section of fence doesn't he have
to have your approval before he puts it on the property line,
otherwise he must put it just inside the property line?

If you say nothing when he erected the fence, you consented to its
placement. Where he puts the fence does not impact your obligations. Either
way, you're in for half.*

I have no idea of where the fence was placed. I did not survey it and
he did not relay this info to me.

I ask these questions because I have no use for fences. The shared
fence is only there because of my neighbors pool. I have no clue as to
if the previous owner of my house had any agreement with the original
pool owner about the initial installation of the fence.

* Here's the drill. You are responsible for your portion of the cost under
the principle of unjust enrichment.

How have I been enriched?

At the first moment you detect such a
project underway, you must announce your opposition. Failing to say
anything, under the principle of "assent by silence," you implicitly conset
to the construction and its attendant costs.

I can't oppose a fence that he uses to enclose his pool. Nor should
I have to pay for a survey to assure he is not encroaching on my
property.

Interestingly, much of contract law can be covered using "the fence" concept
as examples (the rest can be covered by the concepts of "the bull" and "the
pit").

When studying "the fence," you can get off into interesting areas such as
"no notice" (the fence was erected while you slept), "total cost" (you
consented - by silence - to the idea of a fence, but not one made of
polished marble with gold gryphens), and so on.