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GH-36326: [C++] Remove APIs deprecated in v9.0 or earlier (#36675)



### Rationale for this change

General cleanup of C++ APIs deprecated for at least a year.

### What changes are included in this PR?

Removes any APIs annotated with `ARROW_DEPRECATED` and an included version `<=9.0`. In a few cases, tests needed to be removed or slightly altered.

### Are these changes tested?

Yes (covered by existing tests) when applicable.

### Are there any user-facing changes?

Yes

* Closes: #36326

Authored-by: benibus <bpharks@gmx.com>
Signed-off-by: Gang Wu <ustcwg@gmail.com>