Concrete – How to use to patch this gap between brick siding and top of foundation

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 would suggest using composite or PVC lumber in this application. Composite boards are moisture and rot proof. They are made with wood fibers and recycled plastics. Most are paintable so you can match colors. The other choice is PVC boards. Some are paintable, but most should be used in the factory color. If you do paint PVC, use a product that is specifically for plastics, regular latex paints will not adhere properly.

Pressure treated lumber is a good product for this application, however it is difficult to paint. It usually has a high moisture content and must be very dry before attempting to prime and paint. Not a good base for paints, however better for stains.

I'd stick to composites in your situation.