I can not find a single 2 channel device that has HDMI inputs.  That seems to only be available in A/V receivers.  If you want a device that is going to have RCA, USB and HDMI you may need to get a 5.1 or 7.1 receiver.  Seems silly, but I checked all the usual suspects that would have sub $700 equipment (Sony, Onkyo, Yamaha, NAD, Marantz, Denon and Harmon Kardon) and none have what you are looking for. 

The closest I think you can get in a single device is the Onkyo Tx-8050.  It has a built in phono stage and a USB input in front for simple i-pod connectivity.  Yamaha has an external i-pod dock but that is less than ideal since apple just changed it's interconnects and that may pose issues for the future.  It is $299. 

Alternatively, you could purchase an inexpensive integrated amp and an inexpensive outboard DAC.  At the moment, you could get a Yamaha or Cambridge Audio integrated for $350 and Music Direct currently has Marantz's NA7004 network player/outboard DAC on clearance for $399.  It pushes you over budget by $50 but gives you easy ipod plug and play, the ability to attach a computer via USB, coax and optical digital ins.  The integrates have built in phono stages. 

Both of these options still leave you without HDMI connectivity. 

The final option would be to get a 5.1 or 7.1 AV receiver and only attach 2 speakers and ignore the additional functionality.  You should be able to find several dozens of devices that are sub $700 and have USB and HDMI, however a phono stage becomes problematic.  You could buy an external phono stage (easy to find a good one for $200) or again go over budget and look at something like the Marantz SR6007 which is $799. 

Hope this helps.