To shut out the world and worldly distractions. Jesus advised praying in a closet.

I think it helps a person concentrate and focus.Prayers to God do not command a certain position or stance.The important thing is to be sincere and have faith that he hears you.Psalms 65:2 calls Jehovah the Hearer of Prayer.Learn more about the importance of prayer at jw.org.

To avoid distractions, and be able to concentrate on what I am saying, and what God is saying to me.

When a person closes their eyes to pray - the secular world is turned off.

The above allows a deeper concentration preventing all the exterior antics from colliding with our thoughts.

Sometimes, sometimes no.

I do sometimes, and that's to reduce distractions.

Just in case someone is watching...pointing...laughing. GUS STR SYPH

I don't.  Some people do to help them concentrate on what they're doing is my guess. 

Closing your eyes when you focus on anything eliminates external stimulation.

I rarely do.  Sometimes I might at the start of a corporate prayer just to focus my attention.