Classification followed by regression to handle response variable that is usually zero
This sounds entirely reasonable, and the usual name for this structure I have heard for this is just "pipeline" which also applies to other system-feeds-next-system structures - it might also be "machine learning pipeline" or "data processing pipeline".

There are ways to assess performance of a ML pipeline:


You can of course compare the final accuracy or loss value, with the simpler model. Has turning the model into a more complex multi-stage one actually improved things? Sadly nothing is guaranteed, although I would be hopeful in your case initially - in part because you could apply adjustments available to classifier models used to deal with class imbalance issues.
You can decide which part of the pipeline will gain you the most benefit by switching between pipeline-so-far input to each unit and perfect input from the training data. Then you can see how much incremental difference is possible by perfecting that unit in the pipeline.


In your case you have a two stage pipeline, so you can check whether it is worth focusing more effort on the classifier or regression parts by comparing the incremental improvements between:


The unadjusted output of the whole pipeline run end-to-end.
The output of the regression (or zero) assuming that the classifier was perfect.
A perfect score.


Whichever of the two differences gives you the largest difference (2) - (1), or (3) - (2) points at work being most rewarded for working on the classifier or regression stage respectively.

You can see a worked example of this per-stage analysis in Advice for Applying Machine Learning (slides 21, 22), amongst other places.