"Saanko ottaa nämä kaikki omenat. - why only "omena" is in partitiivi but not also "nämä" ja "kaikki"?"
"Omenat" is not partitiivi, it's nominatiivi (plural form).
"Kaikki" and "nämä" are also nominatiivi.
You have to use "nominatiivi" because you refer to something specific, "all those apples", not "some apples" or apples in general. For instance at the market you would say "saanko omenoita?" , if you're not referring to any specific kind of apples, you're simply asking for apples in general. However, if you are asking for some specific apples, you say: "Saanko nämä omenat?" "Can I have THOSE apples?"
By the way, if I'm not mistaking the partitive plural of omena should be "omenoita" but some people say "omenia", I'm not sure if it's a regional use... Maybe Matti (or anyone else) can help us with this :-)