"If someone is renting a room and sharing other common space, such as kitchen, bathroom, living room etc do they have rights since there is no lease or no written documentation whatsoever?"
There is presumably a lease; the fact that it isn't in writing isn't relevant. The tenant still has rights.
"They are using their security for the last months rent."
It's not up to the tenant unilaterally declare what the deposit is used for, so you made a mistake in acceding to this. I would not have allowed it myself.
"This person is refusing to let me into the room to show it unless that person is home."
I don't think NJ law allows for you to enter without the tenant's permission under these circumstances. I suspect you'll have to wait until he-she leaves. (That said, if there's no lock on the door, unless you think she's got a camera set-up in there, there's not much she can do about it if you just open the door and let the prospective tenant take a look inside the room. It's not as though a couple of incidents will make for a big deal even if she wanted to take the issue to court.
"There is no lease, it's just on a verbal agreement."
There is too a lease agreement. :)
"In addition I need to fix the baseboard heater in the room since the fins are broken. The person wonÃ¢â‚¬â„¢t allow me to do anything in there while they are not home. This is not fair and things need to be done."
You can do it after they leave.
"Now my roommate expects me to call them when I pick up my new dog so they can meet at a neutral place not at the house so no issues occur."
Huh? You don't have to consult with the tenant about when you bring your dog home or anything else.
"After the way I have been treated lately I could care less if my dog bits there dogs head off."
Sorry, that's just silly. :)
"The roommate also started complaining about space and wants to get what she paid for."
She's free to allow her dog to roam around if she likes (if you wanted it a different way, then you should have had a written lease agreement setting out prohibitions).
"What would be my best course of action here?"
I'd bide my time for the next couple of weeks.