Topological data analysis is very much in its infancy as a field. Right now there's far more methods available than theorems and the usual "intellectual infrastructure" one might expect from a branch of mathematics/statistics.

My impression is at present people are largely playing with all the ideas in various applied situations, trying to see what kind of inferences the tools allow one to make. This is in contrast to the desire to prove fundamental theorems. Many very basic questions are still open about the tools that persistent homology and its sibling ideas present.

So to answer your questions (1) and (2), yes there are such tools. But given that the field is relatively primordial, the extent you can use the tools to do what you like are largely governed by how lucky you get and how well you can make inferences between topologial computations like homology and geometric intuition.

Topological data analysis is very much in its infancy as a field. Right now there's far more methods available than theorems and the usual "intellectual infrastructure" one might expect from a branch of mathematics/statistics.

My impression is at present people are largely playing with all the ideas in various applied situations, trying to see what kind of inferences the tools allow one to make. This is in contrast to the desire to prove fundamental theorems. Many very basic questions are still open about the tools that persistent homology and its sibling ideas present.

So to answer your questions (1) and (2), yes there are such tools. But given that the field is relatively primordial, the extent you can use the tools to do what you like are largely governed by how lucky you get and how well you can make inferences between topologial computations like homology and geometric intuition.