Walls – What kind of paint should I use to paint a field stone foundation wall

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.

The mortar is crumbling because of moisture seeping through the walls from the outside, not from the paint. That is a common problem with fieldstone foundations. See questions about how to remove crumbled mortar and what kind of mortar to use.

I would be very surprised if the fieldstone itself is actually crumbling. If so, you need a structural engineer that specialized in old foundations. Can you please post a picture? (If the system won't let you, upload it to imgur.com or somewhere else, and someone here can edit the picture into your question.)

Working with the crumbled mortar and peeling paint will probably require protective equipment because of the lead paint. See this related question. Bring in a specialist if you're unsure about how to protect yourself.