I will be putting the utility sink in the empty space between the dryer and the white wall in the picture the washing machine is immediately to the right of the dryer.
Drain utility sink into washer standpipe.
If the washer standpipe is pvc and the sink drain will be pvc of the same size the connection is easy.
Discharging the rinse water into a standpipe keeps the tub clean.
You can install a bracket or wire ties on the back of the sink so the washer drain hose is permanently attached to the sink.
If you can install your washing machine in the garage and next to a utility sink then yes you can let it drain into the sink.
The standpipe assembly keeps.
The original copper pipe was 1 1 4 and the new pvc is 1 1 2.
Buy a t shaped pvc joint cut the standpipe insert the t so that the cross is vertical.
This is the new pvc drain pipe.
You may need a special drain hose depending on the distance of the washer to the sink.
The proper way to drain a washing machine isn t to hook the hose over the edge of a utility sink as many homeowners do but to install a standpipe and secure it to the wall framing.