Lot's to digest here, but I'll take my best shot. You are correct in that most who graduate (usually 80-90 percent) go to cutters for their first tour.
Cutters can generally be divided into three different categories. The first category is the "white hulls," so named because they are painted white. These cutters do the bulk of what the Coast Guard is known to most for: Search and Rescue, drug interdiction, migrant interdiction, law enforcement, defense readiness, and fisheries enforcement. They range in size anywhere from 87 foot long patrol boats which are generally speaking away from home port no more than a week at a time all the way up to 418' National Security Cutters, which can best be likened to Navy Frigates, and they go on deployments up to six months all over the world in support of U.S. interests. The second group is the "black hulls." These are the real workhorses of the Coast Guard. They do nasty work that is mostly involved with maintaining Aids to Navigation (ATON) such as buoys and lighthouses. They are also capable of law enforcement and SAR missions, but it is by no means their primary duty. The bulk of the black hull fleet is made up of 225 foot Seagoing Buoy Tenders and 175 foot Coastal Buoy Tenders, but they are augmented by 140 foot Ice Breaking Tug Boats and 65' Harbor Tugs, which do minor ice breaking and maintenance of small buoys and day boards and the like. Lastly, there is an entire fleet of black hulled river tenders out on the Western Rivers to maintain those ATON and keep commerce flowing there. Officers will pretty much never come into any contact with river tenders of harbor tugs. Lastly, there is the "red hulls." Currently, there are only three, but we are slated to get up to six more by 2030. These ships primary mission is icebreaking; the Healy goes to the Arctic, largely in support of the science mission there, and the Polar Star goes south to the Antarctic annually to clear a path for the yearly shipment to McMurdo station. These two ships have long deployments followed by long drydock/inport periods. Lastly, the Mackinaw is a unique ship stationed on the Great Lakes that primarily breaks ice but also participates in the ATON mission.
As far as travel goes, the bigger the boat, the more you'll see. Smaller boats simply don't go as far, while larger boats are known to pull into foreign ports relatively frequently, especially South America and Asia. There are some pretty amazing travel opportunities in the Coast Guard no matter what career field you go into, and I'll leave that there.
The bases are primarily in the U.S. because the cutters generally always leave from and return to their home port in the United States, but they are certainly big enough to cross oceans and therefore do. Really the only place boats are stationed besides the United States is in Bahrain. There are also cutters outside the Continental US though, to include Hawaii, Alaska, Guam, and Puerto Rico.
Helicopter operations are conducted on all white hulls of a length north of 210' long - so pretty much any major white hull cutter. Additionally, the two Polar Icebreakers are capable of Helo ops as well.
Choice of the major at the Academy correlates to but does not dictate billet. In general, the Mechanical and Electrical Engineers and Naval Architects tend to lean towards student engineering, which is the people that work in the engineroom and with the ships machinery. The other majors tend towards Deck Watch Officer, who are responsible for the safe operation and navigation of the ship. I would say that Civil Engineers are split between the two. You find people interested in flight dispersed everywhere. That said, there are no rules saying that certain majors must take only certain jobs, it just tends to be how interests line up. Any major can do any job.
I am but a cadet and while I do have interesting stories from some of my summers, I will leave sea stories and what it's like to be afloat to those on here who have vastly more experience than I.