Can I tap into the kitchen / dishwasher drain pipe with my washing machine?
Probably not a good idea. Clothes washing machines generally discharge a large volume of water at a high rate, the drain apparatus is built with a standpipe (either inside the wall or outside where you can see it) to accommodate the volume. The standpipe holds the water long enough for it to drain without overflowing. Alternatively, a laundry tub/sink is sometimes used. Your local code enforcement authority might even require a standpipe or tub.