Can you drive over the border to Rhode Island with a Connecticut driver's permit?
no not with a permit but you can with a lisence
Can you legally drive in Massachusetts with a parent in the car if you are 16 and have a valid Connecticut learner's permit?
You can drive in any state that recognizes the permit that you have. For example if you have a Massachusetts learners permit you can drive in CT, as long as you are driving with an adult over the age of 20 and who's had their license for more then four years with out any violations. Source ... http://www.dmv.org/ct-connecticut/drivers-permits.php#Out-of-State_Driving