None of the projectors you listed have any specific advantages for PS3 than many others. That is, they are all excellent projectors.
Since you are a new projector user I would recommend you go with the AX100 projector as it is the most versatile in setup options and therefore, will be the most forgiving if you screw up somehow during the installation.
Do you need HDMI? No, but I wouldn't buy ANY projector without HDMI.
Do you want 1080p compatibility? I don't think that matters at all unless you are buying a 1080p projector. These projectors are not 1080p projectors.
For brightness, quality, and versatility, it's damn tough to beat the AX-100.