What material to use for cosmetic red brick touch ups outside

This all depends on how long the paint has been on the house and what kind of shape it's in. Because brick is a porous material paint adheres very well to it. I think your first step would be going to the local paint store and rent a power-washer. A gas powered power-washer. You are going to need as much psi as possible (at least 3000+) with a pin-hole revolving tip. This will blast away any paint that is loose on the brick.

This should be less damaging to the brick and mortar than sand blasting.

Depending on how successful this was in removing paint dictates your next steps, if most of the paint is gone, then maybe just a wire brush and some paint remover will get the job done. If you've power-washed your whole house and it is still covered in paint, then I would call a contractor and get quotes.

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.