Basement – required to move a washer/dryer from one room to another

There should be no problem locating two gas burning appliances close together as long as you have proper ventilation. The reason your manual says not to put two items is a closet is that they may create excessive heat when used together. As long as you have good spacing, good venting and a source of combustion air, you will be fine.

You want the shortest distance possible, and for everything except perhaps the dryer connection, you want to use rigid metal tubing rather than the flex tube. The rigid tubing will resist the airflow less and collect less lint. The better airflow from rigid tubing and shortest distance will make the dryer more efficient. If too much lint builds up in the vent, you have a fire hazard in addition to longer drying time.

I'd also suggest having the exhaust somewhere you can reach with a small step ladder to make it easier to clean the vent. So I'd run it down through the floor rather than up through the ceiling, if possible.

Finally, when you install the vent opening in the side of your home, make sure it's weather tight. If it's vinyl siding, you can get a removal tool (it's a flat blade with a hook on the end that pulls two pieces apart). Take off a few pieces where you're working, securely attach the vent, seal around the edges (not sure if Tyvek tape is good enough, they make a rubber adhesive membrane for window openings), install a few pieces of J channel like you have around windows and doors, and cut (measure twice) and reinstall the siding. Vinyl siding screws should remain loose (should be able to slide a fingernail underneath) so that it can expand/contract with heat/cold.