If a spouse dies without leaving a Will, is the other spouse entitled to a full joint bank account or does it have to be distributed also among children not on the account?

When someone dies without a Will they are said to have been "intestate". Accordingly, the intetacy or "succession" laws of the state in which the deceased was domiciled at the date of their death will control. These laws vary from state-to-state but typically they provide for a split between the surviving spouse, if any, and the children of the deceased. For example, 1/3 to the surviving spouse and 2/3 to the children.

As for any statute of limitations, that too will differ depending on the jurisdiction. To be sure, check with a local probate attorney.

You should be aware that, intestate succession applies only to property held in solely in a deceased's name. Certain account or assets would pass directly to the beneficiary named on the account. For instance, assets held in joint accounts would pass to the joint owner. 

If a spouse dies intestate, which means without leaving a Will, the surviving spouse inherits the entire estate which would include the bank account you mentioned.  When there is a surviving spouse and no Will, the children inherit nothing.  If there had not been a surviving spouse, the children would inherit equal shares of the estate under the rules of intestate succession.