Joed32 pretty much has you covered on the HR44 question. It still requires a coax connection between it and the satellite, as do all other receivers/mini clients throughout the house. What it adds is WiFi connectivity to your home internet network to give your DirecTV system access to the internet and your computer network. Internet access is used for stuff like downloading on-demand programming, watching you-tube, listening to Pandora, using other web-apps (which I think are shut down right now but supposedly coming back soon). It also talks to your computer network so you can use stuff like the DirecTV iPad/iPhone/Android apps, the Nomad device, and DirecTV2PC.
As far as the install goes I'm pretty sure all HD installs now use SWM, or Single Wire Mutliswitch setups. That means they only need one coax wire to each receiver, and it can be split along the way as long as it isn't split too many times. From your description it should be pretty easy for them to do the install at your home, and they should be able to re-use your current cable runs as long as they are good quality. Since you only want one Genie DVR and two thin clients you will only have 5 tuners, which means they will use a SWM LNB. That means they will only need to run one coax wire from the dish. They will run that single wire to the location where your current 3 way split is. They will replace your current splitter with a splitter of their own (hopefully a 4 way, but a lot of times the techs only use 8 ways), with a higher signal bandwidth rating. They will then install the thin clients and Genie in the rooms you want them in. One of those coax runs will also get a SWM power inserter installed in line on it to power the LNB and multiswitch inside the LNB. They will also most likely install a Wireless DECA adapter (assuming you don't get an HR44) to hook your system up to your home WiFi network.
Hope that helps.