Thank you for providing the details, I've analysed the video in question using FFMpeg.
The 360p original uses the webm container format. The video/audio formats are vp8/vorbis
Code:
Metadata:
encoder : Google
Duration: 00:04:17.72, start: 0.000000, bitrate: 837 kb/s
Stream #0:0: Video: vp8, yuv420p, 636x360, SAR 1:1 DAR 53:30, 23.98 fps, 23.98 tbr, 1k tbn, 1k tbc (default)
Stream #0:1: Audio: vorbis, 44100 Hz, stereo, fltp (default)
Windows doesn't have a native vorbis codec, so I converted the audio to aac, leaving the video 'as-is' and saved to a .mkv.
Stream mapping:
Stream #0:0 -> #0:0 (copy)
Stream #0:1 -> #0:1 (vorbis (native) -> aac (libvo_aacenc))
The resulting .mkv plays perfectly in WMP with none of the problems you describe, so the original video itself isn't the cause of the troubles.
I then tested Download Manager 6.31 on the 360p video. I'm not sure how you got it to output to a .mkv file, for me it created a .mp4 with no options I could find to change that. The .mp4 it created also played perfectly in WMP. Both the audio and video were recoded during the download.
Code:
Metadata:
major_brand : isom
minor_version : 512
compatible_brands: isomiso2avc1mp41
creation_time : 2017-10-13 07:07:43
Duration: 00:04:17.74, start: 0.000000, bitrate: 632 kb/s
Stream #0:0(und): Video: h264 (Main) (avc1 / 0x31637661), yuv420p(tv, bt709), 636x360 [SAR 1:1 DAR 53:30], 503 kb/s, 23.98 fps, 23.98 tbr, 90k tbn, 47.95 tbc (default)
Metadata:
creation_time : 2017-10-13 07:07:43
handler_name : VideoHandler
Stream #0:1(und): Audio: aac (LC) (mp4a / 0x6134706D), 44100 Hz, stereo, fltp, 125 kb/s (default)
Metadata:
creation_time : 2017-10-13 07:08:10
handler_name : SoundHandler
I've done these tests with WMP on two machines, neither of which have additional codecs installed bar those supplied with Windows itself. One was a 32-bit OS, the other 64-bit. TBH, I'm at a loss trying to recreate your problem, it all works fine for me.