You don't need phone lines, unless you want caller ID. Internet connectivity gives you On Demand programming.
You should look at DirecTV's Movers Connection. You will probably get a Genie upgrade, plus Genie mini-clients; suggest you keep at least one of your DVRs (the HR24...) which will be part of the whole home network. you may want to keep more of your DVRs, it depends on how many rooms you are going to equip with DirecTV, and what are your watching/recording patterns. . The Genie will be installed with wireless network connectivity (which will connect all the boxes thru whole home) - if it's an HR44, wireless is built in. If it's an HR34, the installer will install a wireless connection kit.