Speak to an attorney before you do anything. Thats all the advice I can give you on this forum.

Does court consider child’s cognitive skills and communication abilities ?

Generally yes. However, Courts are very determined to make sure that parents get to spend quality time with their children, no matter how advanced...

You need to sit down with local counsel and review your court history and your case. You may need to motion the Court.

I am the Executor of my relative's Will in New York, and was left a house in the Will.. question re transfer of property.

Speak to your attorney to do this properly. Normally everyone waits to make sure that there are no debts or claims on the estate.

You need to speak to an attorney to discuss this matter, asap.

How to I stop my wife from taking children on a trip which might be unsafe or dangerous?

It seems that there is much more than just this trip that is objected to. I caution objecting to this until you speak to counsel and develop a plan...

Does my son need to have his own room/bed for his summer vacation at his father's?

Each child needs their own bed. That being said, an air mattress or mattress on the floor, might be adequate. Please speak to your attorney to...

When the baby is born can the mother of the child put the father name on the birth certificate if he’s not present at the birth?

As counsel says, as long as he acknowledges paternity.

Will my son get in trouble if he leaves his dad at 16 years old?

"Get in trouble" is very broad term. Parents need to follow court orders.

Yes. ...................................