Will a HDMI switch box help me?
No. Video is not your problem and I wouldn't count on HDMI for your audio needs now. Actually, that's a dicey call for anyone nowadays, not just you.
Panasonic SA-HT800V DVD/VHS Home Theater System
1 Component Video Out
1 S-Video Out
1 DVD/VCR Video Out
1 Line Out L/R Audio
1 External Video In
1 Line In L/R Audio
If you look carefully at the specs you posted for the "receiver", all it allows for input is one analog audio input. This will prevent you from any "real" digital sound from anything except it's built-in DVD player. But, you can get some utility out of it for now.
Your TV will allow you to input two HDMI sources which can take care of the DVR and perhaps the DVD recorder's playback. Depending on your particular DVR and DVD recorder, you might be able to get by with the component inputs as there is no technical reason to "need" HDMI for HD, but some manufactures force that requirement . RTFM to see what they allow.
Use the TV as your source selector. So much for video.
You can "get by" by running the DVD recorders red/white analog audio to the corresponding input on the TV. Likewise, you should do the same for the HD-DVR, but you won't get the 5.1 digital sound from either. ...more on this later.
And, many people (myself included ) have no problems using the TV asthe video selector and the receiver as the audio selector. for one thing, it allows me to play the TV with it's built-in speakers when I don't want to use the big system. Ever hear Sponge Bob in 5.1? Trust me, you don't need to.
In any event, you will need to run the TV's red/white analog audio output to that one lonely red/white analog audio input on your receiver. When you use the receiver to select this input, whatever is playing on the TV will be sent to the receiver ...in stereo.
The Philips DVD recorder doesn't even appear to offer a digital audio output so stereo may the the best it's capable of delivering.
With that stereo audio, you should be able to select some sort of quasi-multi-channel sound emulation, such as Dolby Pro Logic (DPL). Again. RTFM for more.
To get true digital sound, you'll need to run a digital connection from the satellite receiver directly to the receivers digital input, which isn't an option at this time since it doesn't have one.
Oh, as for recording, I have no idea, Perhaps another can offer some advice but I'm outta ideas.
...you might really want to consider a receiver with the required inputs as your next investment. Decent ones can be had for < $200 nowadays.
good luck and I hope this helps somewhat.