If someone is keeping a vegetarian/milk household, and then they acquire a dog that eats meat based dog food products, what potential Kashruth issues might arise? Would there be difference in Kashruth between wet or dry dog food?
Dog food is typically not Kosher. That's OK. One's dog does not need to be fed Kosher (compare with Shemos 22:30). However, that does not exempt the requirement to keep the food away from Kosher dishes. The exact details of "away" are a bit beyond this answer, but it would have to be kept away just like any other non-Kosher food, even though you are allowed to use it.
The exception is anything that one cannot derive benefit from. Feeding one's own dog is a benefit. So that means milk and meat mixtures are forbidden provided the combination is in such a way that it makes them biblically prohibited (see the link for details), as well as anything Chametz on the week of Passover - those are two common problems faced by dog owners in terms of sourcing food.
The dog food is not Kosher, and should therefore not be prepared in an area or with vessels that you want to keep Kosher.
Be especially careful, if anything needs to be warmed up.
If it gets on your dishes or counter-top you should probably speak to your local Rav and ask what to do, though if it is cold and you completely remove it, it should not make too big a problem (I would still ask).
Best solution would probably be to cover the area you are going to pour over, or designate a separate dog food area.
You'll need to make sure that it's not meat and milk together, because it might have the issur of hano'oh (prohibition of having benefit) that applies to it. For a much more in depth answer, check this out: http://www.torah.org/advanced/weekly-halacha/5763/tetzaveh.html
i had originally accepted an answer stating to prepare this non Kosher dogfood in a separate area away from the area that i prepare my food. After consulting with my local Sephardi Rav, he said that these are the issues to be concerned with:
If the dogfood is never cooked, and always remained cold, then it is unlikely it will absorb into anything. Heat is required to make something absorb. So there isn't need to worry about if this food lands on any countertops, plates, etc. There is worry if bits land into my food, so i should prepare the dog's food at a separate time.
When cleaning, one should be careful to use lots of soap to render any leftover food particles pagul, and to clean any dishes that came into contact with it using cold water only. These concepts are laid forth in the Shulchan Arukh. Being any stricter than this is a chumra, and while beneficial, is not necessary.
The dog food should not have any mixture of meat or dairy in the dog food itself.