If you plan to take your dog into places where he is illegal, you may be in for a nasty surprise.
Dogs are most definitely not allowed on trails in the national parks. The one exception I know of is in Washington State, where they can be taken on the portions of the PCT within Mt. Rainier NP and North Cascades NP if kept on leash at all times. If you're caught with a dog on the PCT anywhere else in a national park (such as Yosemite, Sequoia or Lassen), you'll get a ticket with big fine and will be escorted to the nearest exit. The same is true with the several California state parks through which the PCT runs--absolutely no dogs allowed on trails. National monuments--it depends on the individual monument; check beforehand. There are wilderness rangers out there. There are also some hikers who are so anti-dog (often with good reason, such as having been bitten or had their food stolen or packs urinated on) that they just might turn you in to a ranger.
The best way to do this would be if you have someone to pick him up and care for him at the national park boundaries and deliver him to you afterwards. That's what PCT hikers do who have taken dogs. There have been a few cases of people hiking with service dogs. I just hope they were legal, because if this privilege is abused it will soon lead to the banning of service animals everywhere.
Those 25-30 miles per day hikes that PCT hikers routinely do are very, very hard on dogs. There are also portions of the trail in volcanic areas where the dog's paddy paws will be chewed to bits. I won't take my dog on the PCT in Oregon's Three Sisters Wilderness for that reason alone. Friends who have done this stretch say they would never take their dogs there again! There are also many stretches of the PCT where it's a long distance between water sources, so you'll have to carry extra for both you and your dog. On hot days, even with plenty of water, your dog will be much more subject to heat stroke than a human.
To be successful, you'll need to hike your dog's hike, not your own. Pay very close attention to his condition and don't hesitate to leave the trail if he's having problems. I had to abort two much-anticipated trips (both in Wyoming's Wind Rivers) due to my dog's becoming ill--once two days after starting on the trail, the second because he was diagnosed as needing surgery a few weeks before the trip. You also need to keep your dog leashed most of the time (especially in camp or where there are a lot of people around) and to scoop his p00p (treat it the same as human p00p, by burying it in a cathole).
Please talk to your veterinarian about your plans! A class in dog first aid would be an excellent idea.
Some articles that might help:
I have no knowledge of South America, but to start your research you might want to find out the requirements each country has for importing a dog. http://www.letsgopets.com/inttravel.php#s_america
has a list of requirements but I have no idea how up to date it is.