For example, if I skipped the first chest for Magic level 1 or Melee level 1, and went directly for chest level 2, do I still get the benefit out of level 1? Does that mean I can save seals doing so? Or does each chest increment the level by 1?
While you didn't take the previous chests of the same skill, the next ones cost the sum of every chest, and you will get the benefit for every previous chest too.
In other words, if chest1 costs 10 and you didn't take it, the next chest will cost, for example, 30, and will cost 20 after taking the first chest.
So dont waste time waiting and saving seals if you are planning to take a skill upgrade.