With HSK lists, if I start with HSK 1, then go to HSK 2, 3, 4, 5, and then 6 ... each new list doesn't contain words from the previous lists, right? So, HSK 1 has words that only HSK 1 has, then 2 adds to that, then 3 adds to that, right?

So, if I want to add all HSK words, then I would just add all lists and add words as I go?

Even if they had some words on more than one list (which I don't think so) that wouldn't make a difference.
Skritter knows what words you already learned and only adds actually new ones. (you will also notice that if you're looking at a list you haven't studied words you already know will be marked green).

Yes, that's how it works with the (new) HSK lists.

I'm currently going through these lists as well, but find it harder to learn due to less context compared to learning from a textbook list.