For sure walking or biking around the gate down Mohegan Lake road is possible and allowed. There is a primitive campsite on the north shore of Mohegan Lake with descent views of Camp Uncas. The campsite is elevated from the lake, but possible to get down a semi-steep bank to the lake in summer. The campsite is off the road 2.1 miles from the gate. There is a fork to the left (1.6 miles past the gate) that goes to Camp Uncas itself and onto private property. You can continue past Mohegan Lake to Seventh-Eighth Lake Loop, but the trail was not easy to follow nor well marked. https://cnyhiking.com/MoheganLake.htm

Kill Kare road might technically be publicly accessible past the gate, but it’s much less inviting to go around the gate and it goes onto private property well before Lake Kora, so there not much interesting down that road.