Ok, this is something I’ve done a lot of so you may get more than you need and I’ll almost certianly not cover everthing, it’s just to difficult.
As has been said MakeMKV simply allows you to copy the show, selecting the tracks you need.
There’s no economies of size or conversion for added compatibility with playback devices etc.
But first the subtitles you had a problem with.
Some playback devices can’t overlay subtitles on the video which causes Plex to have to transcode the show. If the show video is h264 then Plex will likely use hardware transcoding and while the quality might not be as good as the original it’s usual pretty good. But MPEG2 (what appears to be in your show) is usually not available for hardware transcoding so the CPU must be used to decode it which often leads to problems like what your seeing.
The TS-251+ is not going to do well with CPU transcoding so you need to supply it with something that it can use hardware transcoding for, h264 is about the best choice.
Anyway, back to the subtitles.
It sounds like you have found that a movie you were watching has forced subtitles and so your adding the full subtitles track to work around it.
That’s usually not necessary, you’re best to try and work out which track has the forced subtitles and copy only that one.
Subtitle tracks can have a mixture normal subtitles and forced subtitles but movies usually don’t use that, they have an additional subtitles track that holds only the forced subtitles. You can tell if a subtitles track is a forced subtitles track because it will have very few entries compared to normal subtitles, it’s usually obvious.
You can check this by coping all the subtitle tracks for the language you want in MakeMKV and then using something like the mediainfo program to check how many entries each subtitle track has after it’s been copied.
So the you know what subtitle track to select when you encode with, say, handbrake.
Next you need to concern yourself with the audio.
Simplest thing is to select just the language audio you need in MakeMKV and if it isn’t Dolby Digital less than or equal to 5.1 channels and less than or equal to a bitrate of 384 kbs then convert it to this when encoding with handbrake.
It gets a bit more complicated if you have HD audio and you have a player that can do audio passthrough to a receiver but that’s out of scope here.
Finally the video.
We won’t go into selecting video codec, lets just say h264 is the most compatible and use that.
Start Handbrake and select a preset that’s close to what you have in terms of video resolution and farmrate.
Select Matroska as Format and in the video tab select “Same as source” for framerate and “Constant framerate”.
I prefer CRF encodes myself, so still in the video tab select a CRF value of 20 (you can play with this later), “Constant quality” will already be selected.
Proceed to the audio tab and select the highest quality audio track (I don’t have a video to load into Handbrake right now so this might not be accurate).
If it’s an AC-3 track that meets the characteristics above select AC-3 passthru.
If not then select AC-3 and the characteristics above for the converted audio.
Proceed to the subtitle tab and select and add the subtitle track you identified above as the forced subtitle track, you can select “Burn in” for this track to merge it into the video.
Set the output location and video output name and start the encode.
This should produce a video that’s virtually indistinguishable from the original in terms of quality but somewhat smaller and with the most broad compatibility with playback devices that you can get along with subtitles that are only what you want merged into the video frames.
I have had problems with burning in subtitles like one or two not displaying as long as they should (flashing up not long enough to read) and also problems with positioning if the video resolution is also changed but mostly this should work ok.
Hopefully I haven’t missed the point and this (or some parts of it) is useful, 
Ian
