You wouldn't be able to do the Precipice Trail anyway. It is also closed because of nesting peregrine falcons.
The Beehive Trail is pretty short. Perhaps you can combine it with something else. Remember that you don't have to end where you start because you can use the free Island Explorer buses to either take you back to the trailhead where you started or back to Bar Harbor.
The Maple Springs to Great Slide and up the back of Parkman is a great hike. I personally think the view from the top of Parkman Mountain is even better than the view from the top of Cadillac. What you need is a really good map. You can't hike from the free map that the Park gives out. Just not enough detail. Map Adventures map of Acadia National Park is the one to get. Perhaps you can wait until you get here and buy that map from either the Park's Visitor Center or Sherman's Bookstore in Bar Harbor.
With a good map it will all become clear about where the trail heads are and which trail connects with which other one. The map I recommend has distances and elevation and parking areas. In addition to the ones you've researched, a nice hike is up the Perpendicular Trail on Mansell Mountain, outside of Southwest Harbor, and down the Razorback Trail. The mountains on that side of the island are very different (and less popular) than the ones on the Bar Harbor side of the island.
The trail head for Beech Cliff is at the southern end of Echo Lake. Big parking lot there. You can also access the trails there by driving to the very end of Beech Hill Road, where there is a parking lot. That lot will put you near the top of the cliff.