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Pretty sure what you're looking for is something called FlexView: http://www22.verizon.com/home/fiostv/ondemand/ . That's mainly the On Demand section of it, but with everything else that you mentioned, such as streaming Live TV, don't think FiOS has that yet but I have a feeling they are working on something. They do have Remote DVR and Guide, though which is mainly used to schedule recordings and manage them remotely.
It may be the case. I am not a FiOS customer, so I naturally do not have access to this application nor do I have the bandwidth available at home to even use such applications. I would imagine they would have content available based on what you subscribe to, though.