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.
There is disagreement about what constitutes an acceptable mathematical proof. E.g., is proof by contradiction a proof?
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.
No, it doesn't, but as a matter of fact there has been disagreement among mathematicians on those issues. See e.g. https://en.wikipedia.org/wiki/Brouwer%E2%80%93Hilbert_contro....
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.