Your contractor is goofy that's what he is.
The drain IS NOT constructed properly but is not lost. I would compact the soil in the bottom of the drain trough and pour some concrete in there and finish it off smooth. The fact that it isn't tied to the walls is only a minor concern because there is more than enough natural irregularity in the walls that the base casting will lock itself to the walls anyway and won't likely go anywhere.
You could then either hot-mop or cold-tar the base and walls which will seal the junctures and eliminate any cavitating under the trough.
I would allow the bottom of the drain tube to be about two inches above the finished bottom of the trough. This would collect any solids that may get into the trough over time and could be cleaned from time to time. That's better than allowing any water-flow to carry solids into the drain pipe that might settle mid-pipe and cause an obstruction over time. This way the drain tube can't be filled with mud that is being washed from the trough during use.
OK, now that door.
This too is obviously a fiasco but the floor can be ground down slightly. I would strike a straight line equal to the inside of the door and begin grinding the high spots from that line outward and along a path under the door seal. A sharp edge at the point of just inside where the door sits won't hurt anything and the outside can then be ground off to allow water to run away from the door.
Don't let the door company tell you the door can't be adjusted downward a little, that's nonsense, they can all be adjusted from time to time. Of course it won't seal like it is but somebody with a little imagination can make all this work and work properly.
I wouldn't be recommending that concrete guy to any of my friends in the future either.
The only other option would be to have the concrete contractor start cutting and removing concrete in both areas and start over and I think he would try to buy-you-off before he went to that trouble for free.