Is this the same as the War Memorial building?
No. The War Memorial Recreation Center is an indoor pool and gym facility at 1640 Cambridge Street; the park listed here is the outdoor Joan Lorentz Park in front of the Main Library on Broadway.
Is there an entry fee at Library Park (Joan Lorentz Park)?
Cambridge has no parking sticker program for its parks and no municipal swim beach; Fresh Pond is a drinking-water reservoir where swimming and wading are prohibited. No pass or permit required.
Can I bring my dog to Library Park (Joan Lorentz Park)?
Cambridge requires dogs to be leashed, on a leash no longer than six feet, whenever off the owner's premises and not under direct control in a designated off-leash area. That's the town-wide default; watch for posted signs here.