Already some great replies and most of what Ill say follows along the same lines.
Food: A goat on the farm here gets a good flake of dairy quality alfalfa (20%+ protein) per day. Which in weight is 5-6 lbs. We like first cutting the best as it has a more substantial stem then later cuttings, which in turn means more fiber. Fiber is what slows down a goats digestion process so that they are able to absorb more of the nutrients in what they eat. A working goat can actually lose weight on a mostly leaf and less stem later cutting regardless of protein because it passes through their system to fast. A bright green soft poo is a sign of this.
Translating this out in the wild is a bit tricky as there are so many factors. As mentioned, time of year, vegetation available and so on. But keep this in mind, you would be very hard pressed to find anything in the wilds nearly as high of protein as a dairy quality alfalfa. Good thing is, anywhere around 12%-14% is more then enough for a working goat as long as they are getting enough of it. Grasses, leaves, shrubs should be more then enough if in the right quantity. The less a goat has to do the less quality of feed it would need. Meaning, you will not maintain a healthy goat out in a desert environment packing heavy loads for days on end. So you will need to maybe do some research on what kinda food stuffs will be in the area you wanna spend your extended stay (good time to look up and get familiar with toxic / poisonous plants.) If its not going to be enough then you will need to take in a food source for the goat. Now as mentioned above, a large goat should get roughly 5 lbs a day.Making a 50 lbs. bag of alfalfa pellets lasting just 10 days. Store bought pelleted alfalfa is at most 15% protein. But on the flip side, they are not wasting any of it like they do eating outta a feeder on the farm.
Dog Food: yes they will eat it and no, you should never purposely feed it to them. Goats are not meat eaters and there are meat by products in most dog foods. And although I would highly doubt a goat would be at risk for "mad goat disease lol " the feeding of cow by products back to cows, is how mad cow was spread. Just kinda food for thought and no not serious about the mad goat
Water: A daily water source is a must on extended stays. Goats can go a few days without water once in a while, but not a good idea to do this repeatedly over an extended stay. The warmer the area, the more water they will need of course.
Shelter: Goats are very hardy animals except when it comes to rain and wind. A mix of these are a major source of pneumonia in goats. We feed with no cover over the feeders year round, the goats may get a bit damp either from light rain or snow but when it gets to heavy, they run for their barns and will stay there until the rain / snow stops or is very light. Wind isnt so much an issue by itself. So a dry goat is a healthy goat is the best way to look at it. So tarps are fine so long as they keep them dry, thats including the ground they will be laying on. If its wet, bedding it down with ever green branches would work not only as a dry place to lay on but a mid night snack as well

African goats were mentioned but none of the ADGA accepted dairy goat breeds are from Africa and have many hundreds of years as domesticated livestock under their pedigrees and for the most part they have had barns for shelter for just as long. So although not fragile, a happy goat is a.... happy goat
Intended use on the extended stay: Now if you are just looking for a companion to take on a nice long walk about to carry a bed roll and supplies, you could do no better then a goat. Regardless of size id try to keep the pack to under 30 lbs and have the goat conditioned before I set off. Id tried to find my animals sweet spot for weight. Meaning walking for several hours before it needed to lay down for a break. Being new to the packing world, myself, these are some of the things Ill be looking forward to doing myself. Im not much of a stop and go kinda hiker. If there is a goal I wanna get to, then I wanna get there in good time. Not counting breath taking views or geographic marvels.
So as asked, more information about where and when you want to do your stay would be very helpful to give more info. And from an experience point of view, you would do better to pay heed to those with it then those like myself who have yet to earn it. But when it comes to goat care / health, I like to think of myself as pretty darn good. And that is from experience
