Should Coax Splitter Be Grounded?

Grounding through the splitter is not recomended as a lightning strike or sytem power surge can essentially melt the components of a splitter together and allow the surge to hit your devices. Not having a ground will NOT cause any damage to your devices unless you get a surge of some kind.

Do coax splitters need to be grounded?

No.... the splitter should not be connected to ground in a receptacle like that. The system gets grounded at the point of entry. The splitter next to the outlet box as you have it is probably the first access point to the cable system that you have control over.

Does a coax splitter weaken the signal?

A cable splitter WILL result in a degradation of the signal, even if the other ports are unused. One thing you can do is to add terminator caps to each unused port. They are supposed to reduce the degradation. Note that cheaper cable splitters will actually have a different amount of signal loss for each port.