Here is how to reproduce since I can't explain this one very well either.
These steps go in correspondence with the screenshots.
1. Put a minecart onto a rail.
2. Spawn something. A skeleton is what I used. Push him into the minecart.
3. Usually, it will go off the rails by itself, since when the mob got in it pushes it in one direction. If it doesn't, make the cart go off the track.
4. Right-click to get in the minecart. You can get in it while the skeleton or whatever you spawned is still inside.
5. Either wait a few seconds or hit the mob to make it get out. It should now sink into the ground.
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Can confirm.

It seems as if the whole "Get into the minecart while something else is in it" may be an entire separate bug.

Something similar occurs with me. A few different minecarts w/ or w/out mobs suddenly sink deep(not visible anymore) into the ground when not on a rail track. I was at the moment making some other carts go in powered loops.

this happens every time you break a minecart with a mob in it. Also on a saddled pig this fixed itself by riding around for a while.

Still in 1.5.1

Is this still a concern in the current Minecraft version 1.7.4 / Launcher version 1.3.8 or later? If so, please update the affected versions in order to best aid Mojang ensuring bugs are still valid in the latest releases/pre-releases.

Can no longer reproduce in 1.8 (used powered Activator rails, and derailing minecarts then breaking them to make the mob get out. You can no longer ride a minecart that a mob is currently riding.)

Seems like a duplicate of MC-18969.