I'm assuming its the one on the right as you sit on the bike, and I am assuming that it loosens anti-clockwise.
Yes and yes. You'll need a spanner on both ends. Galling is microscopic welding and tearing of metal under pressure. In some cases the threads of the axle and axle nut gall, ruining the axle and nut. If you turn the nut anticlockwise and it gets harder to turn, stop. An expert craftsman might be able to put a penetrating lubricant or heat on the nut and work it loose without damage. A less than expert craftsman will twist it so hard that it rips the threads out requiring replacement of the axle, or get things so jammed that the axle must be cut to get it out.
By the way, how loose is your chain? If it is still within the measurement in your owner's manual, don't tighten it. A bit loose is better than a bit tight.
When you do get the axle out, here are two tips (but the axle doesn't need to come out for chain adjustment)--
--Reinstall it from the right. All the various parts line up better for installation with the axle inserted from the right, and the direction makes no functional difference.
--Put a small dab of an antiseize product on the threads...Copaslip or something similar...then tighten the nut to only about 80 Newton-meters, tight, but not quite as tight as the specification in the manual.