Raleigh R300 Touring, has a 135 rear drop out. Using an RSX triple front crank. What do I choose for BB size, road or MTN? Using RSX front derailleur and rear derailleur. Is the BB size based on crank or rear dropout measure?

I would base it on shell width, 68mm or 73mm..etc

For the RSX triple, the stock BB length is 118mm with a Shimano cartridge BB (BB-UN##) and will give you a chainline of aprox. 45mm. You could go to a 122mm Shimano cartridge and get aprox. 48mm. Don't worry too much about getting the perfect chainline, a few mm doesn't matter. With a multispeed bike, the effect is small, and may just be noticeable in the crosschain combos. As long as the chainrings clears the chainstays you're good. As you have seen already, Shimano's FD have a pretty wide usable range.

Other manufacturers cartridge BB are similar in specs, e.g. Tange. If you go to older loose spindle BB, check the spindle for asymmetrical L/R lengths. Those spindle were all over the place regardling L/R.

The correct length will be determined by the crankset design, but as already mentioned can be adjusted a little.

The chainline at the rear will have some variation depending on frame spec., # of speed, freewheel or cassette, hub configuration, etc.

I will go with 118. With friction shifting should be ok