Walls – Should I do something about these cracks in the basement wall

Three inch gaps seem quite large - even for a 300 year old wall.

You just need to use a good mortar mix (sand and cement) on the narrower gaps.

Clean out all the old, lose material first to make sure that you get good adhesion and then push the mortar right into the gaps as far as it will go. Work on a small area (1 m²) at a time.

Adding pea gravel to give the mix more "body" might work on the wider ones.

However, I'd get a professional to have a look at the wall first to make sure you're not going to have to redo this work in a couple of years time. If the foundations have gone it might need something more substantial work doing.

Is the wall part of a house or just a garden wall? If it's the latter then dismantling and rebuilding might be the best long term option.

The pocket door that is dragging the floor is the best indicator perhaps of what is going on. If it is original to the house means one thing, if it is a newer install as in not original means something different.

If painting the wall covers the crack you are asking about, tells me they are very small and there is most likely no concern. Old houses move, Plaster walls don't like to move and when they do, you will get settlement cracks, which it sounds like what you have here. The again if the pocket door is an add on or part of a remodel, that will affect the wall. Still no concern. The door that is dragging the can be from a few possibilities. The header supporting the track failed. I don't think so, your wall you asked about would have worse cracks. The track may have become detached from the header and the door dropped. (poor workmanship)

It may be good to post another question with pictures about the pocket door. That appears to me to be more of an issue than the cracks in the wall.