There are a couple things wrong with your connection diagram.
First, the cable in the upper right appears to be a PS3 Composite AV cable (yellow = composite analog video; red/white = left/right i.e. stereo analog audio). While you could connect the red & white plugs to the red & white audio input jacks on the game recorder and get audio to pass, trying to connect the yellow composite video plug to any of the (3) red/blue/green component video inputs on the game recorder will not give you the result you are looking for. If it even picks up a video signal, the color would be way off. Furthermore, composite cables are only capable of carrying a max resolution of 480i (SD). If you want to be able to record in HD then you have to use a cable that can carry HD resolutions (Component/VGA/DVI/HDMI/DP). For this particular recorder, the input has to be Component. Luckily, they make PS3 Component AV cables and the PS3 will allow HD video to be passed via. Component when playing games (though not when watching movies), so this part of your problem could be resolved by replacing the PS3 Composite AV cable with a PS3 Component AV cable. However, this might not be the best solution for this half of your problem due to your next issue...
I could be mistaken, but I'm pretty sure the cable you show in the lower lefthand corner is NOT a Component to HDMI cable. If somebody sold you a cable and told you that that it could convert Component to HDMI then they scammed you. While it is physically possible to make a cable that has Component connectors on one end and an HDMI connector on the other, the signals that are carried by cables with these types of connectors are not compatible with each other. Component cables carry an analog signal for use with analog circuits. HDMI cables carry a digital signal for use with digital processors. It is possible to convert a signal from one form to another, but to do so requires an additional converter. AFAIK, no cable exists that has one of these converters embeded into it. They must be purchased separately and they are not cheap. If your monitor has Component inputs on it then you would simply use a Component cable to connect the game recorder component passthru to your monitor's component inputs. If your monitor does not have Component inputs but does have a VGA input then you could buy a Component to VGA converter (Amazon currently has a Monoprice converter on sale for ~$50) and use Component cables to connect the recorder's Component passthru to the converter then connect the converter to the VGA input on your monitor using a standard VGA cable. If your monitor does not even have a VGA input, (i.e. HDMI input only) then you are either going to have to buy a Component to HDMI converter, an HDMI splitter & HDMI to Component converter, or a new monitor that has both Component and HDMI inputs. (Note: if you go the HDMI splitter & HDMI to Component converter route then either the splitter or the converter will have to have EDID managing capabilities to get past the PS3's HDCP when using HDMI).
In response to your last question, the PS3 can only send out an AV signal via one output at a time (either HDMI or an AV breakout cable). It won't pass a signal to the game recorder via PS3 Component AV cable while also passing a signal over its HDMI output. So, if you want to use HDMI to connect the PS3 to the monitor then you will have to use an HDMI splitter and then run one HDMI cable from the splitter to your monitor while running another HDMI cable from the spliter to the HDMI to Component converter as mentioned above.
Edited by HockeyoAJB - 2/19/14 at 1:36pm