Sorry, this one seems to have slipped under the radar.
You can learn each parameter that is exposed by the plugin to Reaper's automation system to MIDI CC or note. The easiest way to get there is from the plugin's gui. Clicking the button labeled "Param" in the window header gives you some options for the last parameter you wiggled with the mouse, one of them is "Learn". It also appears on envelope lanes and in the window that pops up when you click on the env button of the track.
If you have many parameters to attach, it might be a good idea to use the corresponding action (FX: set MIDI learn for last touched parameter). That one is super-new, now that I try it the Learn window doesn't tell me which parameter is actually addressed by the current learn
There are checkboxes in the Learn window to define the kind of control that is learned (relative or absolute) and whether the control is global or only happening when the track or the plugin UI has focus, so you can use the same CC number on multiple parameters.
When done save that instance of Sylenth as a FX chain from the FX chain window of the track (you can also save single plugs of the track as a chain by highlighting it and choose "save selected FX as chain" from the FX menu. That will save all learned parameters and track control knobs.
Seeing which CC already has been learned to which parameter is a pain in Reaper. You can only find out one by one by visiting the learn window.
What I don't get is the "the rotary knobs and sliders seem to be bound to completely random parameters throughout Reaper and my VST's" part. Might be Sylenth's own learn system.
Ah, yes the transport buttons would be learned from within the action window, like when you assign keyboard shortcuts.
*waving DarkStar as he tears past with high speed while I type