Some are, some are not. It pays to check this out. Sometimes fees are assessed on an annual instead of a per use basis.
Fees are most common with privately owned parks.
Dog fights can and do occur. The most successful dog parks are the ones with an active user group. An active users group can reduce the risk of dog fights significantly. Most dogs are not aggressive because they are not on their own territory. The dogs that use it most successfully are dogs that were socialized very young and had good experiences as they visited.