You should get what makes you happy. In choosing try to picture it with your other choices to make sure they all work together. When I read that you're considering a limestone BS, I immediately thought of a
very traditional kitchen.

I wouldn't use it for countertop (porous and soft), but since it's vertical where it doesn't get wear, and especially since it's not in the major staining area behind the stove, I think it could be a decent option. You should be able to seal it to protect from water and other splashes for occasional cleaning.

I think it would be fine for a backsplash. I have honed limestone floor tiles in my master bath, and rustic French limestone floor tile in my kitchen. Both have been sealed with a good sealer and I don't have any issues with stains as of yet.

My kitchen counter tops are carrara marble (similar to limestone) and I don't have any staining on them either, and just a little etching here and there but it is not really noticeable.

I have seen limestone in magazines for counters and I think it would be fine there too, as long as you are okay with patina.