There's plenty of disagreement among people who actually think about what it means to prove or disprove a mathematical claim (which is a tiny, tiny minority of mathematicians). See finitism, intuitionism, etc.
There is unlikely to be an agreed-upon mechanism for determining progress in philosophy for the obvious reason that what constitutes philosophical progress is itself a difficult and rather important philosophical question.
The groups you've mentioned set forth their axioms and accepted logic and construct their math accordingly. There's no disagreement here, as there's no "disagreement" between Euclid and Lobachevsky.
Yes, it is certainly a proof (in a most commonly-used logic systems)
There may be some logic systems where proof by contradiction is not a proof. The existence of such logic systems does not mean there is any disagreement between mathematicians.
There is unlikely to be an agreed-upon mechanism for determining progress in philosophy for the obvious reason that what constitutes philosophical progress is itself a difficult and rather important philosophical question.