1) Cubes is not a good plan. That's definitely a lower bound. Put four balls together, their centers making a lovely tetrahedron, and see how much space is between them. This might be a good start on increasing the number. There are other problems at the walls, however.
2) You'll love this. If you number the balls, you might get a different outcome. I explain:
a1) Throw in 10 balls. b1) Find #1 and remove it.
a2) Throw in 10 balls. b2) Find #2 and remove it.
a3) Throw in 10 balls. b3) Find #3 and remove it.
an) Throw in 10 balls. bn) Find #n and remove it.
Continue this process. This leads to no balls at all. Every ball eventually will be removed. :-)