A/V setup
Sounds like a pretty good "starter" system.
You definately need to run all audio through the receiver in order to take advantage of your new speakers, surround sound, etc. Due to the upscalling features of the Yamaha, it is probably desirable to let it do all the video switching as well. You will have a single HDMI cable from the Yamaha to the display and wont have to worry about changing display modes on the TV.
For the cable box, yes, a DVI to HDMI cable or component video cable will be fine for video. Use a digital coax or optical cable for the audio.
If you have more then 2 HDMI or DVI devices, then you will need to connect them directly to the TV's additional inputs, use a component video connection, or purchase an HDMI switch box.
Good luck with the receiver setup. Blue Jeans cable and impact acoustics are good sources for all of those cables.