For natural stone seal, it is not very recommended to use chemicals to clean it because the chemicals might harm the stone surface. There is a very popular way in China to clean the seal. I use kneadable eraser. This can help clean the small details on the carving. I think this way is quite good.
I’ve never used different inks with the same stone before but I don’t imagine it’d be a very big deal. You may want to clean it as best you can but if there’s a little of the previous color left over I don’t think it’d print with it if the new ink is going directly over it. Just my thoughts, might want to try experimenting with it on scrap paper but I highly doubt it’d harm the stone in any way.
I sort of agree with you! Just try to keep the seal clean as possible. Of course it is impossible to absolutely clean off the ink. And yes, you can use different ink colours. Using scrap paper is an idea but this might damage the carving. Maybe a toothpick would be more helpful.