Premature Grandson

My grandson who is now 3.8 years old was born premature.  He was born 7 months premature
and was under the care of a neo-natologist for a month.  Meaning he was kept
in the hospital for a month while the mother has gone home.  He was born
weighing 1 kilogram or 2.2 lbs.  Both of his lungs collapsed at birth and was
given surfaktan to survive.  He had brain hemorrhage  but disappeared after a
few months.  He was cleared from retinopathy, his ears were okay and
miraculously, he is now 3 years old and 8 months and is doing well and
healthy.
Our problem is his behavior.  He is so hyper active, throws things around, scatters them all over the floor, does not know how to gather them inspite of your letting him and helping him,  he loves to color, though not so good, can recognize his alphabets, primary colors, can recite numbers 1 to 15.
He is starting to write some letters but our problem is when no one is with him, he starts coloring/writing on the walls, TV screen.  We have reprimanded him regarding this and have even punished him but to no avail.
My question is, Is this normal?  It really bothers me as my son, never did this when he was young, my niece who I have reared till 10 never did this too.  Even us at children, never did this.  (Me and 2 of my other siblings).  My sister's daughter and her grandchildren do not write on walls too.  Neither does my brother's children.  My daughter-in-law said, in their family, almost all of them did this.  Is this normaL?  Does this run in the family, hereditary or comes from genes?
What can I do to stop this?
Thanks.

[AnnaE]

No one has answered you! This happened to me, too. Sorry.

I think you should talk to the pediatrician first. Have you already done this?

Marianne