Water – Is mortar waterproof

It is not so much about mortar as it is about the working process. You should clean out the old crumbling mortar to make place for the new one. Remove the old mortar so that all parts are firm and nothing is crumbling - you can even scratch out a bit of the old mortar. The very important part is to wash the wall properly with water so that all parts hold firm and there is no dust - otherwise the new mortar would just fall out. Than press the new mortar in the joints and fill them, make them smooth on surface.

As for the mortar the lime content should not be important for the lifetime, but you can add a bit for better workability. Generally there should be no or very little lime in a stone wall. The new mortar should consist of 1 part of portland cement and 2-3 parts of sand.

If you do it properly (well prepared surface, good mortar), the new mortar should not crumble or dust for tens of years.

A personal experience: it is a very time consuming monotonous work, but the result can be very good if done with care.

I wouldn't be too concerned unless there's mortar that is raised above the level of the grout - which you've already said there isn't.

Scraping the higher sections to give a reasonable depth of grout would be as far as I'd go.

A lot of adhesives these days are sold as tile cement and grout, though they do tend to be coarser than "pure" grout, so they are basically the same material. However, a pure grout does give a smoother, "cleaner" line.

Floor tile grout also tends to be coarser - I suspect to be able to cope with the higher degree of wear and tear it will receive compared to wall tile grout.