After flushing, water level in bowl fills and then slowly drops.

you have a blockage maybe outside in your shores

2 possibilities.1. The water level is set higher the the overflow tube. This will cause the water to run into the overflow continuously. You should be able to see this. 2. The large seal - bottom center of the tank - is leaking. This will cause the water level to slowly drop after a fill. The water tank will start to run again after flushing - BUT NOT CONTINUOUSLY. You should see if there is a small amount of water flowing from the rim to the bowl 15 seconds after the water has stopped running after a flush.

The small black tube that goes into the overflow is used to fill the bowl after flushing. Would check to make sure that water is coming out of that tube at full force or if its coming out at all.

As long as there is no water leaking from the tank to bowl connection, you are OK. You would know if it was leaking here, because you would have water dripping out onto the floor.

Verify that the water level in your tank is below the top of your new flush valve's overflow tube. If the water level comes to the top, it will overflow into the toilet bowl. This can be easily fixed with an adjustment to the fill valve.

If the water is not overflowing into this tube, then the only way you can lose water is through the flapper. You said you changed the flush valve, which would have come with a new flapper attached to it, but maybe buy a new flapper and see what happens???

Most likely, you do not have a good seal on the flapper valve at the bottom of the tank. To confirm, try this:
Turn off water and flush toilet. Then lift up the flapper valve and put a light coating of vaseline on the mating surface where it contacts the toilet tank when closed.
Turn water back on and see if it continues misbehaving. If that fixes it, it was a bad seal. Usually that is all it takes.

Check the fill tube in your toilet one easy way to see if it's leaking is get food coloring and pour in your tank to color the water this makes it easier to see and squeeze the tube in the tank when it's full if you see the colored water running down the middle of it you have a winner! if not you can look around for it sometimes the plastic under the flapper gets cracked too

First off,If the air vent was clogged the toilet would back up and overflow and when it did flush, it would make a gurgling sound.
secondly the water level in the bowl is not important as long as it covers the opening,The water is only there to keep out sewer gases so they don't enter your home and the shape of the trap will always maintain the proper amount of water needed to prevent this.
The previous water level may have been due to a partial clog that has since been removed.

First, fyi, adjusting the knob? in the tank, or the fill level will not increase the amount of water in the bowl. The bowl fills to a certain point, and then as more water is added it starts to go over the built in trap, and down the drain. When you flush the toilet, all the water in the tank dumps into the bowl, and the existing water is pushed over the trap and down the drain. All of this being said it sounds like you may either have an internal crack, between the bowl and trap; you may have a crack that is draining under your toilet, (water spots?), or you have a more serious plumbing issue, improper venting, which is causing a vacuum, and pulling the water out. Hope this helps.

If very little water is used on initial flush, The bowl is functioning as a tank for second thorough flush. The high water level cannot stay high in the bowl while in motion or you would have a mess ion start and stops.

Make a small mark with a pencil or marker that can be erased of were your water line is in your bowl. Pour some water from a bucket into the bowl SLOWLY and see if the water line rises or drops back down to your original mark. If it drop back to the original mark, then that is the way the toilet was made. You won't be able to do anything about it. If it rises and stays at a higher level, then you have a problem with the fill tube in the tank. You will need to call a Pro!