Drill into plastic with Dad's drill?

There is no harm in drilling into plastic. It is soft. The drill will try to move around on the surface, so hold it steady and do not push hard to start. Let it make a small dip in the plastic that will hold it in place before you push harder.

Use the drill bit that suits the size of hole you want. 

While drilling through plastic can be done using any power drill, it's important to have the correct drill bits so you don't chip or crack the material. ... The most effective drill bits for plastics are spur-point bits, also known as dowel bits. They are specifically angled to make smooth cuts in the material.

Yes, it's okay to use a drill for that. Use whatever size bit you need to get the size of hole you want. However, some plastic doesn't drill very well. It's too brittle and it shatters easily. The ice cube tray will probably be fine. The bucket has a better chance of breaking around the hole as you drill. Putting some tape masking tape, where you want to drill may help, as would putting a piece of wood on the other side from where you're drilling, and the smaller the bit the less likely it is to break the plastic.

Yes, it will drill very easily into plastic, so you need to be sure to hold it steady. Size will depend what the holes are for.

OF course that is OK.  No idea what size to use because we don't know the purpose of the holes.