Welcome to out little niche here on AVS. Hopefully we will be able to help you along the road to HD.
1. Regarding repairing the JVC: I live in Sumter so really can't help you there, sorry.
2. Regarding the purchase of a HDTV. If you are wedded to Sony and that particular model and have done your homework then... good luck! I might suggest you spend some time here at AVS reading forums especially the direcview one, the one on rear projection and the four HDTV related ones.
3. IMO, the issue of DVI/HDMI is something to get a handle on as is Firewire/5C. Regarding DVI/HDMI capable DirecTV receivers, I am not sure there are many choices yet. For now, besides the Sony SAT HD200 you mentioned is it's twin, the Zenith SAT-520 HDTV (both are reportedly made by Zenith but there are some differences). Other models are due later this year.
4. Local HD broadcasts are available OTA or from Time Warner Cable. Using the OTA receiver built into the DirecTV STB coupled with a VHF/UHF antenna (for our area) works well for many in our area. Hopefully it would for you too! Time Warner also offers most of the OTA HD channels. The station and TWC must first agree to the carriage and so there can be a delay from the station turn-on date to it's being on cable. DT stations currently on TWC are CBS, ABC, and PBS; an agreement with FOX is being worked; and NBC and WB are not yet on the air.
I somewhat understand your feelings for TiVo having used TWC's Explorer 8000 DVR for a couple of months (TiVo is reportedly much more mature and superior in operation at this time). But your talking about spending $600 or more now for a receiver then another $1000 (est) when the HD TiVo comes out. FWIW, digital cable including one HD box starts at about $60/mo.
I mention cable because of cost and bandwidth (channels). TWC might be a cheap way to get HD now and let things get sorted out a bit though not many want to pay for both cable and a dbs. A number of new HD channels have been proposed (see the Program Synopsis listed here
near the end). It is unlikely that DirecTV will be able to carry them all due to limited bandwidth. Cable has the bandwidth advantage but again no one knows yet which channels they will carry either. One caution I might make is the current HD cable box, the SA3100HD. does not work well with all 4:3 HD capable sets due to its lack of aspect ratio control (next model should be better).