Apart from dogs that are completely or mostly furless (chinese crested) etc then a dogs coat should do for protection. I have Pyenean Mountain Dogs (Great Pyrenese in the US). They have very thick fur coats with two layers and are relatively immune to cold and wet. Ok they are a bit on the extreme side and wont actually get out of the shallow hole they have dug in the lawn to seek their kennel or tarp unless it is raining heavilly, muddy or below about -4 degrees centigrade. Their coat is ridiculous in comparison to shorter coated breeds but its not cold or wet you have to worry about (especially cold) but heat. Particularly with flat faced breeds. Pekinese, boxers etc. They can die of heat stoke very very quickly even in the UK. Shade is more important from sun than anything else.
Shelter needs for dogs can therefore be basically the same as ours (tarp to shelter under, something to lie on if the ground is wet or very cold) with the addition of a secure ground anchor (looks like a giant metal corkscrew) or tree to tie them to. I find my dogs are happy enough in any conditions as long as I am with them. They would not want left behind.
The other thing to take is a collapsable water bowl unless you dont mind sharing your cooking pot!, although mine have been trained to drink from taps, streams from bottles and using water pistols (I have a small son). They will happilly drink from muddy puddles with no ill effects (I frisked them but have never found any purification tablets, milk bags or pumps so either something funny is going on or they hide them well <img src="/images/graemlins/wink.gif" alt="" />). Food wise they usually get dog food and view "people food" or leftovers as a massive treat. In a real emergency stuff the dog food they will happily eat what I eat (and weirdly will enjoy it more because they are sharing with me - talk about easily pleased)
My FAK is specifically geared for use on dogs as well as me and contains some veterinary meds.
Unfortunaely mine expect food to be provided. However I used to have a lurcher. They should almost be part of your emergency equipment - as long as you like eating rabbit.
With my pack along my family do not need firearms! They would kill or die to protect us. Least I can do is make preparations for them to leave Dodge with the rest of us.