yes recheck on the ground. your actually suppose to be sitting on the bike, and have a friend adjust the chain with you in full gear. If its ona stand, it is being pushed against the swingarm making the tightness of the chain change.
That nut is 36 mm I think. Socket on 1/2 drive flex bar works good and retorque to 78 foot pounds. I use a cheapo floor jack to lift the back wheel to adjust the chain and then its easy to spin some lube on. But like someone else mentioned the tension should be checked with a full load on bike. I use a clip like retainer that I can reuse. I hate trying to bend the cotter pins so I can get them in and out.
When adjusting the chain make sure you have your weight on the seat, when you check it. You want about 1.5-1.75 inches of play gixxers need a loose chain for them to handle right. The trick on gixxers is to have it loose to about 2 inchs so when you tighten up the axle nut it should tighten to 1.5
Also its a good idea to buy stands, if you cant afford them. Try buying handy stands 99 bucks you get both.
I've been there. I didn't have any clue about a single thing. I was so lost when I bought my 2002 600, that I probably shouldn't of even owned the bike in the first place. But look in your bag. Its goes look like any orndinary tool. Its pretty much a handle attached to what would be a cut out of your rear axel nut.