Can one assume juice to be kosher even without a heksher on the container because fruit is assumed to be kosher and does not need a heksher? What about produce that has been cut, such as watermellon slices?

100% pure fruit juice is kosher even without a hechsher but you must first check the ingredients to determine that there are no problematic additives. In Israel there must be hechsher so that you know tithes were taken etc. Cut fresh fruit does not need a hechsher because it is cut with a clean knife used for this purpose only.