Yes Harmony will do all that. Get the Elite and you should be able to program the buttons on the button to control lights and outlets. I have not switched my "Pro" remote for the Elite yet, but it is on my project list, so I can add some lighting functions to the remote.
Harmony does integrate with HE via community driver from @ogiewon. Ex. I have a button on my RA2 keypad that talks to HE/Harmony/Lutron to run the "Good Morning" routine of a few lights and turning the TV on the wife's favorite news channel. The same should be able to be done via the remote. When remote button X is pressed do Y. Again I have not done it that way yet, but it's on the short list.
FYI be sure to reprogram the remote so that it sends the IR signal via the remote (not the hub) to the TV and all other signals via the hub. This is done via the Harmony software (on PC I'm not an app guy when it comes to setup). The hub will live in the AV room, and the remote will send signals to it for device control (IR and RF). The remote does not need line of sight to the hub. The have installed at least 25 remotes with the exact configuration you describe.
