It is a rechargeable battery and comes with a USB-C charging cable, but no adapter. 

Yes. I have the same lens and adaptor on my Sony a7ll and it works beautifully. The m42 is also known as the 'universal' mount as other 3rd party camera and lenses also used it. I own 6 other K&F e-mount adaptors to different lens makers (ltm, MD, CF/CD, Nikon) and they all are well built with a nice snug fit. Highly recommended. 

This projector doesn't have a built-in battery, but they provided a power adapter in the package. 

Please note: our M42 adapter ring could not be suitable for Pentax Takumar Super Takumar Carl Zeiss M42 series（aperture pin does not fully depress, small tabs sticking out), as ours depresses the aperture pin all the way down

 

I actually have not received this adapter yet. It's shipping from China. Don't think that was disclosed in the ad. However, I do not believe this is for the Pentax Q body rather for digital and film cameras that take the Pentax K mount in its many variations. So I'm expecting to be able to mount my Leica M mount lenses on my digital and film Pentax K mount cameras, no autofocus or relay of exposure information, only to be used with manual focus and exposure.