Love the hat!
And yes, it's a fantastic feeling, isn't it!
No need to rest your skeins, no, but you should wash them to 'set the twist'.
After washing and rinsing (a little vinegar in the penultimate rinse is said to rebalance the pH), I do snap them between my wrists and thwack them against a wall, seems to shake the fibres into what will be a nice relaxed state once the skein is dried.
Do not ever hang a weight on the drying skein unless you are seeking a specific outcome and know what you are doing. In general, if you weight it while it dries, you will get an unpredictable outcome when you wash the final knitted item as the twist will go back to its natural shape when wetted
By all means block the finished garment if you want. I usually just shape it while wet and dry flat, but do block lace shawls.
The other finishing that skeins may sometimes want is fulling/felting. If it's a woollen yarn that could pill, you can felt or full (not-quite-felt) the yarn to make it more cohesive. Or you can full the finished cloth, which is what they do with jumpers on Shetland - by stretching the jumper on a jumper board and setting it outside to face the elements!
I'm still learning about fulling, and have just started to weave, so I don't know much more than that yet.