If I read your post right, zozoman I do not believe everything you want to do will be possible with the equipment you listed.
1. Going from DVI > HDMI > DVI will be no problem. All you're doing is chaning the connector type. It does not affect the signal.
2. The "splitter" you linked to is not a splitter at all. It is a 2 in 1 out switch. As far as I can tell it will not work the other direction (1 in 2 out). It is made to switch between two sources and send to a single monitor.
In order to accomplish what you want you would need an active 1 in 2 out HDMI splitter like
THIS. Unfortunately those aren't as cheap as switches.
3. You will need to switch resolutions in order to maintaing the best possible picture quality. Digital displays always operate best at their native resolution. It is possible that your projector may not recognize the resolution that you are sending to your monitor as it is not a standard NTSC resolution. What projector are you using?
BTW: I love Newegg and buy all of my PC parts from them. However,
Monoprice.com has better prices on the adapters and adapter cables.