On my visits, which admittedly were a few years ago, even with the 3- and 4-day tickets you had to choose one park for the first day, then the other for the second, but then could hop on the third and, with the 4-day ticket, on the fourth day.
Also, if you stay in one of the 3 Disney-owned hotels, a length of stay Monorail ticket is included for each registered hotel guest. As the Disney resort monorail is part of the JR network, the JR passes that allow unlimited rides on any JR trains for a week or more are valid for the TDR monorail as well, but of course it wouldn't be the best use for the JR pass, value-wise.
Yep, the "Magic Passports" now include park-hopping, though only on the third and fourth day (I've corrected that in the piece.) I chose the two-day passport, so I didn't actually get to do this, though.
The local train to the Disney resort is at one extreme of the station, so there's a good chance you will have trouble finding the right platform. If that happens and you get confused, there are a few information kiosks that have english speaking hosts.