When parent dies without will can the adult children take possession of the deceased parent's property and assets in Florida?

That depends on whether your parent was married at the time of death and if yes, whether the surviving spouse is also your parent. If the surviving spouse is not also your parent then the estate will be shared 50/50 with the surviving spouse getting half and the surviving children by a first wife sharing the other half. If the surviving spouse is also your parent then the surviving spouse gets 100%. If there is no surviving spouse the children get 100%.

It is likely the estate will need to be probated. You should seek advice from an attorney who specializes in probate matters and who can provide up to date information.
Yes if they go through probate. There is an intestacy law that specifies distribution. Without going through probate, the state could claim the property. Consult an attorney in Florida.
Yes if they go through probate. There is an intestacy law that specifies distribution. Without going through probate, the state could claim the property. Consult an attorney in Florida.