Chasing Incoming Timecode
Most of the information you need on Reaper capabilities are spelled out from section 21.6 Syncing to an external device onwards in the Reaper manual. It includes syncing Reaper to external MTC , LTC , SPP(Song Position Pointer) and ASIO positioning protocol.
Sending Timecode
Sending MTC can happen in three ways.
First, you can set up a MIDI port to send MTC in the Preferences page Midi Devices, which is further detailed on the same section of the manual as mentioned above.
Second you can place a SMPTE LTC/MTC Timecode Generator item in the timeline. This command can be found in the Insert menu. You can configure the item, which can be trimmed out as far as one wants, by accessing the source properties of the item, accessible in the context menu of item or via the default shortcut CTRL+F2(Windows).
Third, you can use the JS plugin vVIMTC (virtual Vertical Interleave MIDI Timecode Generator) which you can pick up in this thread.
vVIMTC: Timecode when stopped or paused - Cockos Incorporated Forums
That JS plugin will send a highly configurable MTC timecode out even when your transport is stopped, making it rather handy for external video players.