"When is the first year someone can get into these leagues, and how does one normally get invited?"
Occasionally you'll see a kid go right after HS ends in June. But this is still pretty rare, and it's limited to the lower-level leagues. Also most of the cases I've seen have involved kids playing close to home (and therefore sleeping in their own beds).
But these days, most D1 programs try to place just about everyone who hasn't graduated into a summer program. Exceptions are made for pitchers who threw a ton in spring and are deemed to need some time off; for injured guys on rehab; and for academically marginal guys who need to go to summer school -- though often the approach for them is to try to place them nearby so they can do both.
At the Cape level, the players are pretty much building their pro draft resumes, though of course everyone seeks also to improve by facing tough competition. At the lower level leagues, you'll see guys who redshirted on rehab, you'll see freshmen who didn't play much sent to get playing time and experience, etc. For those leagues just one cut below the Cape -- Valley, Northwoods, Alaskan, Coastal Plains and others who jockey for bragging rights -- you'll see more upper classmen, more studs, guys who just missed the Cape, etc.
High level players often learn within their first month on campus where they'll be heading the next summer. Others the coaches have to use their connections to find a summer home for. The Cape spots get filled pretty early on but there are always guys who bow out late so there are going to be openings late as well. Also there are opportunities to fill in as needed, e.g., in June for guys playing in Omaha, or in August after some guys might sign from the June draft and head off to pro ball.
Commitments are often confirmed with written contracts. In some cases you can say, "I'll come here unless I get a Cape offer" if the team will agree to it. But 99% of the time, your college coach has a plan for you and you don't have a say in the decision.