My guess would also be a
Farragut class destroyer based on that slightly angular rear deck house and the placement of the masts and funnels. The only other class this made me think of would have been a
Providence class cruiser, but they lack that steeply angled hull as you approach the bow. I actually have a brother who served on a member of this class, the USS
William V. Pratt (DDG-44). To address your other questions:
NSMike hit this, but yes, they were American, originally built and commissioned in the late '50s/early '60s. The 10 ships in this class were all decommissioned in the late '80s/early '90s and sold for scrap.
Those look to be practice missiles (hence the blue color) mocking up the RIM-2
Terrier surface-to-air missile. Depending on the version, the
Terrier was either a beam-riding missile or utilized semi-active radar homing and usually featured a conventional warhead, although one version, the RIM-2D, featured a 1kT nuclear warhead.
Yes, those are
AN/
SPG-55 radars, which were the tracking/illumination radar for the
Terriers.
Now? Help ensure you get a close shave

. Then? Their primary mission was to help defend a carrier battle group from attack by bombers armed with air-to-surface missiles. The
Terriers were their primary armament and served this purpose. They also featured a 5" gun to engage surface targets as well as a launcher for the RUR-5 ASROC antisubmarine rocket and torpedo tubes for (at the time the photo was taken) Mark 44 or Mark 46 torpedoes. Make no bones, about it though, their primary purpose was defending the airspace around the carrier.
[Edited 2015-11-11 07:36:46]