Mini DisplayPort was created in 2008.

If that Mac has DisplayPort for video out, then yes it does.

Yes if the PC has DisplayPort out and the projector VGA in.

If the HP has a full size DisplayPort out and the projector is VGA in. Yes

It's unlikely that HDMI based video will work with a DisplayPort input. Although DisplayPort can be connected to an HDMI input with a passive adaptor, DisplayPort does not support broadcast color space such as YUV. HDMI can accept and display the RGB format used by DisplayPort but DisplayPort cannot handle YUV which is typically output from a Bluray player.

DisplayPort is an emerging standard and during the time it is being implemented, there will be numerous practical adaptations to the capabilities of DisplayPort. For that reason, this answer may not be valid in the future. Indeed, it is likely that there are already products that can handle YUV so the only sure way to find out if it works is to try it.