I've checked every forum I could, but I haven't been able to get the specific help I need. I really hope that someone here can take on my challenge:

I recently bought a Panasonic DMR-ES35V DVD/VHS Recorder, and am having trouble incorporating in to my system. I bought it simply to transfer VHS tapes to DVD (which I can figure out how to do) and to transfer shows from my DirecTV Tivo playlist to DVD (which I cannot figure out how to do). I already have a DVD player, so I don't need to use this as my DVD player (though I would if it made sense).

The DVD/VHS Recorder has 2 RCA Inputs, 1 Component Output, and 2 RCA Outputs.

I also have:

DirecTV HD DVR (HR10-250), which has an HDMI connection to my TV, 1 Component out connection, 1 RCA output, and 2 Satellite feeds in. I do not currently have an RF In or Out.

I also have a Sony Dream Theater System, which I am now using for both my surround sound and my DVD player. It has a Component output and 2 RCA Inputs.

My TV is a Sony HDTV, with a Component Input, an HDMI Input, and 2 RCA Inputs.

I can't seem to get the TV to play through the new DVD recorder (which I assume I need to do if I'm going to record from my DirecTV TiVo to my DVD recorder).

Can someone please recommend the ideal solution for me? Your help would be greatly appreciated.

Hopefully the DirecTV box also has an SVHS output, which would be the preferred input for the recorder. The Panny has an SVHS input. It's then just a matter of setting the Sat box for the correct output format (4:3) for recording via the SVHS.

Your recorder should be monitored on the TV with the component connection. That's the only option that will give you max quality for playback and monitoring.
If your TV doesn't allow inputting both HDMI and/or component video, you have a problem. Hopefully they are separate inputs. Since you also want to monitor your DVD player via component video, you already have a conflict and maybe the recorder should be used for playback as well. Or, get a audio receiver with component video switching.