Trev - 59 mins ago »
Only the 4k Firestick needs an external supply (So Amazon says), so unless your stuff is 4k (I haven't checked) ......
Chromecast comes with a seperate power supply but it has a usb socket in it.
According to spec it uses 2W at 5V so projector port unlikely to supply enough current. Presumably the projector needs power anyway. So powering either a Chromecast or 4K firestick shouldn't be a major issue.
Only option to view a 5000T output on the projector is either a long enough HDMI cable or a pair of baluns.
eg
The OP would only need to use the ir return if the projector is used in a different room to the 5000T.