A gas washer -- I don't think so. You need a 20 amp GFCI receptacle for the washer. Rerouting the gas for the dryer through the crawl space should not be a big deal but I would get a certified gas installer(or plumber) to do the work. You will need permits for the electrical and plumbing.
Coastal South Carolina
The washing machine/ dryer connections should be a somewhat easy task. Just make sure it's done right. The dishwasher can be supplied from the hot water stubout under kitchen sink and the drain can tie into the tubular or the disposal easily. Make sure you have plenty of access to the front of the furnace and the water heater for service and replacement purposes and also for safety from combustable building material. An alteration such as this should be permitted and be installed to current code. Make sure if you're doing this yourself to check on current requirements. If hiring a pro, make sure they have a good reputation and that you feel comfortable that they know what they're doing.