[popcornmix]

* In general, Kodi needs a huge amount of time just to start playing (I can reproduce on OS X, so this is not really OMXPlayer or Pi specific): 3-4 seconds on OS X and 5-7 on a Raspberry Pi. [1]
A mkv with the very same streams (converted from AVI with Mkvtoolnix) loads instantly on both.

Normally you see this effect with avi files with a broken index. The whole file gets parsed creating a temporary index.
But I'm not seeing this, and the file opens quickly.

I notice your log has "CFileCache:rocess - Hit eof" - have you forced file cache to be enabled for local files? Can you try disabling that?

* Apart from that, OMXPlayer seems a bit confused with its frame rate. Please note that I have an override in as.xml to force a refresh rate of 23.976 Hz for 23.976 fps content.

Why the as.xml override? The Pi should handle 24 and 23.976 rates correctly without an override.
I do see that omxplayer is detecting double the framerate - I'll look into that.
I suspect that setting "Adjust display refresh rate to match video" to "On start/stop" rather than "Always" will avoid this issue (as will disabling omxplayer acceleration).