1. Do any THROUGH trains operate via Belmont Park ? (The map shows this line continuing on to reconnect with the Babylon Line.)

2. Do ALL Montauk trains operate via the Babylon Branch (Laurenton-Amityville) ?

3. If so, what trains, if any, use the connecting line between the Ronkonkoma Branch and the Montauk Branch (Bethpage-Babylon)?

1. Do any THROUGH trains operate via Belmont Park ? (The map shows this line continuing on to reconnect with the Babylon Line.)

To the best of my knowledge, no. I believe this is just a spur to the racetrack.

2. Do ALL Montauk trains operate via the Babylon Branch (Laurenton-Amityville)?

No. I took a peek at the current weekday timetable online for the Montauk branch, and it shows one train stopping at Mineola; to do so it must run via Hicksville. However, I think it used to be that trains not scheduled to stop between Babylon and Jamaica could be routed via either line, and this may still be the case.

3. If so, what trains, if any, use the connecting line between the Ronkonkoma Branch and the Montauk Branch (Bethpage-Babylon)?

Again, any of them could, as far as I know.

I rode all of the Long Island except the branches to Long Beach, Far Rockaway, and Hunterspoint Ave. I liked the lines out east best--Greenport, Montauk, and Port Jeff. Oyster Bay and Port Washington were all right. The others I rode only for the mileage and that was about the only reward--especially the Flatbush line.

2. Do ALL Montauk trains operate via the Babylon Branch (Laurenton-Amityville)?
Yes, unless there's a stop in Mineola. THis is because the Main Line (via Mineola) is a 2-track at grade branch. Babylon is grade-separated. Also, they do not enjoy trains going through Mineola w/o stopping (Pt.Jeff +Ronky trans don't stop Rush hours b/c there's no way around the sta.)

3. If so, what trains, if any, use the connecting line between the Ronkonkoma Branch and the Montauk Branch (Bethpage-Babylon)?
The Montauk trains that stop at Mineola