MediaMonkey should change the metadata embedded in the file, if you edit the various metadata fields. You will see this happen, as MediaMonkey will give a green progress bar as it writes the modified file to disk. You should also see the file modification date change. (see 'aha' below...)
It may take Twonky a while to pick up metadata changes, depending on the library update setting. If necessary, you can force a library rescan.
It looks like the M&J media players should be DLNA compliant. Therefore, you can use any DLNA-compliant media client to select media to send to it; you are not obliged to use the one they suggest (in theory). MediaMonkey running on a PC is a DLNA-compliant DMC, and can stream audio to a UPnP/DLNA renderer, so you could use MM as your media client:
Tools/Options/Player/Choose Player/UPnP-DLNA Renderer
select the DMR device you want to play to (your M&J player).
MM can access a DLNA Digital Media Server (e.g. Twonky), but that's not how I use it; I point MM to my MyCloud media drive, and let it index it itself (aha: that may be where you're going wrong: if you access the Twonky DMS using MM, rather than getting MM to create its own library by scanning the disk, it probably cannot modify the metadata tags stored in the Twonky database).
MM can also provide a DLNA DMS...
Admittedly, you may not want to have to have your PC running to play music, in which case, you might look to a smartphone or small tablet to be the DMC (Digital Media Controller). I use BubbleUPnP running on Android devices, or MusicBee running on a small Windows 8.1 tablet (bought secondhand for £25, intended to run MM, but it doesn't have enough RAM to run my 65k track MM library).
What metadata appears on the M&J streamer screen is a different matter; that may be determined by the 'Receiver Type' Twonky has selected/M&J has informed Twonky. Since the M&J player is new, I don't expect it to appear in the list of possible Media Receivers that Twonky knows about, so you may have to play with trying other receiver types to see which works best with the M&J.