Q1. Given that my TV only supports 1080i do I gain anything out of using HDMI connections rather than component connections? I don't mind buying more HDMI cables but unless I don't gain any performance in sound or video.
from what I can tell from the manual, your receiver only has component inputs (3) and one component output. Most people do not really see much of a difference between HDMI and component under actual viewing conditions.
Q2. In order to take full advantage of the 5.1 surround sound do I have to use certain types of connections (i.e. coaxial)?
Yes, you need either Toslink optical cables or coaxial audio cables from your sources into the receiver. The typical red-white audio cable is stereo only
Q3. Should I connect everything into my receiver and just have one cable to my TV?
You can connect the video directly to the TV and only route the Dolby digital audio to the receiver, or you can connect everything to the receiver, video and audio, and let the receiver be the main switch for which source material you will be viewing. This does mean the receiver will be on all the time the TV is on, but they are made for that and if you set the a/v receiver remote up properly it can control the TV and the receiver from one remote. This is usually the best option for a family where others will use the system when you are not there to switch all the sources/remotes about to actually watch something on TV.