1. The children lay down on their stomachs.

2. The children lay down on their belleies.

3. The children lay down on their fronts.
(Which expressions are right?)

4. Lay on your right side on the floor.
5. Lay on your left side on the floor.
6. Lay on your front side on the floor.
7. Lay on your back side on the floor.
(Can we use like this? Do we have to delete 'side'?)

First, learn "lie" versus "lay".

http://www.quickanddirtytips.com/education/grammar/lay-versus-lie

"stomachs" and "bellies" mean the same thing. "Belly" is informal and not as polite.

No, you do not have to delete 'side". We do say "left side", etc. "Backside" usually means one's buttocks, though. "Back" and "front" don't require "side", but it's not wrong.

I corrected the errors. Thank you.

1. The children lay down on their stomachs.

2. The children lay down on their belleies.

3. The children lay down on their fronts.
(Which expressions are right?)

4. Lie on your right side on the floor.
5. Lie on your left side on the floor.
6. Lie on your front side on the floor.
7. Lie on your back side on the floor.

1. "The children lay down on their stomachs" is the correct expression.

2. "The children lay down on their bellies" is also a correct expression.

3. "The children lay down on their fronts" is not commonly used. It is more common to use "stomachs" or "bellies" to refer to the front of the body in this context.

Regarding the second set of sentences:

4. "Lay on your right side on the floor" is a correct expression.

5. "Lay on your left side on the floor" is also correct.

6. "Lay on your front side on the floor" is not commonly used. It is more common to use "lay on your front" or "lay on your stomach" to indicate lying face-down.

7. "Lay on your back side on the floor" is not commonly used either. It is more common to say "lay on your back" to indicate lying face-up.

In general, when describing body positions, it is not necessary to include the word "side" unless you want to specify which side of the body to be on.