| Hey Needsinfo:
I see no major issues with you moving the laundry over to the bathroom plumbing as you suggest.
The biggest issue I can tell will be that you will need to relocate the electrical wire for the dryer (or gas piping if you have a gas dryer). You will also need to run the dryer exhaust to a reasonable location.
In terms of the washing machine you want to install a single lever washing machine valve...see image. These are great and should be used to shutoff the water to the washing machine after each use.
Ideally, the pipe to connect into will be a 2" drain pipe. Is the sink drain at the back basement area 1.5" or 2"? Let me know.
You can connect the drain, water and vent all together in a washing machine BOX...see images. These install into the wall and hide all the ugly pipes if you want.
You will want to install a PTRAP with a standpipe about 30 inches tall and you will want to run an 1.5" vent up to the vent you mentioned above the bathroom.
The piping diagram below shows a typical piping arrangement for a washing machine box with a lavatory drain next to it(on right). In your case, you may want to eliminate the piping for the lavatory, but the piping for the washing machine stays the same.
Let me know if you need more...
MARK | 
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