Wort oxygenation really isn't a big issue for the typical kit brewer - most dry yeast is pre-conditioned with the lipids needed for the reproduction phase. Also, most of the directions are for a partial boil and a top-off with water - and if the top-off water hasn't been boiled, it should contain enough dissolved oxygen to do what you need. I suspect that's why oxygenation isn't often mentioned in kit directions - they're trying to keep things as simple as possible. Oxygenation becomes more important when doing full wort boils (because the oxygen is driven off during the boil), and critical when using yeast in liquid form (that includes re-used yeasts that were originally packaged dry).
That said, oxygenation is always good practice to do. I do partial boils and add chilled spring water - I shake the jugs vigorously and pour them through a strainer into my fermenting bucket. Then I pour the chilled wort through the strainer into the bucket at a quick rate, splashing it into there on purpose. This has proven adequate for my purposes, even when using liquid yeasts (with a starter) I've never had a batch under-attenuate.
If you have the air pump and stone, you can certainly use it - I know lots of other folks do. I'm not sure about the filter question.