I have a mini doxie who often "coughs" a lot after drinking water and recently, he's been eating lots of grass and then later, vomiting.

what could this be a sign of?

could be anything from very minor to life thretening. Please take yuor dog to the vet, there is no way to diagnose this on the internet.

My spitz also coughs after drinking water. Could be that water got into the nose.

Dogs eat grass when they feel that they are getting not enough nutrients from their food. Unchewed grass will lead to vomiting. Hope you can gain some insight from this site: Why do Dogs Eat Grass?

I have a 7 year old dachsund whose name is Trixie and when she gets excited or pants too much she starts coughing. I took her to the vet and mentioned it and they did not do a thing about it. I do believe she could have allergies like I do. We try to keep home cleaned of dust excetera. You are right after they drink she gets an air bubble and coughs herself in a tizzy. If you find out otherwise please let me know.