Why do the ice cubes freeze at the bottom of my ice maker? Should I use a rubber or other bucket?

Answer 1

Probably it's a problem of keeping the correct temperature.
Another thing is pressure. The state that water is in (liquid vs solid), depends also on pressure.
I don't know the specifics of your ice maker, but if you have a lot of ice cubes, then the ones at the bottom might be under more pressure because of the weight of the other cubes and thus melt a bit and re-freeze.
A cool experiment to get this point is te following.

Put a thich ice-column horizontally. Tie a weight with thin wire from the middle of the column. The weight will apply through the wire enough pressure so that the ice starts melting at the points where it is touching the wire. The wire will start moving donwards, cutting through and entering into the ice column.

Since the excess pressure stops being applied at the part of the column from which the wire has already passed, the water will freeze again (due to the heat capacity of the surrounding ice) and seal the part that was cut.
This way, if you stop at the right moment, you will have inserted the wire in the middle of the column. If you let it, the wire will go through the ice column, without cutting it into two...

Answer 2

I...don't quite understand the rubber or other bucket thing...
But the ice cubes are freezing in the bottom of your ice maker because your freezer isn't holding temp consistently. That would cause the cubes to melt a bit and then freeze back together.