Question about staff mods, ilvl and skills

As far as I understood maxroll's staff mods page an item with ilvl <= 11 can only have lvl 1 and lvl 6 skills. So if I use the cube on an low quality item to get a normal one with ilvl 1 it can never have a higher skill than a lvl 6 skill on it. Is that correct?

In theory yes....
But I think that the staff mods are not re-rolled when you cube a low quality item to get a normal one with ilvl1

Side node, that is sort of related.

Did you know that there's a way to get a rare shrunken head with +2 to Necromancer skill levels and +3 to Raise Skeleton (and +X to Skeleton Mastery)? It's not normally possible (it can't drop), it can only be done with the cube recipe to re-roll rares (using 6 Perfect Skulls) on elite shrunken heads, preferrably Succubus Skull or Bloodlord Skull for their high qlvl.

It's not normally possible because +2 to Necromancer skill levels on a shrunken head requires alvl >= 50, but for it to get +X to Raise Skeleton, it needs to have an ilvl <= 24. Incompatible conditions. However, the cube recipe for re-rolling rares with Perfect Skulls reduces the item's ilvl with each re-roll, and after enough re-rolls the ilvl will have dropped below 24. And, if you're doing it on a rare of a base item with a high enough qlvl, the qlvl will set an effective lower threshold for possible affixes, allowing high level affixes to be rolled despite the ilvl having dropped low enough to roll a level 1 skill like Raise Skeleton.

This is of course extremely difficult to hit.

Wasn't it also true that if you "fix" low quality Necro head with RS or SM that it will always yield RS or SM mod as well after the reroll or something like that?

I know some guy who was buying chipped heads in-game at the rate of 3 for 1 Jah and then upping their quality because he was trying to hit 3/3 for the ultimate Summoner Necro build and he mentioned something how game works along those lines I wrote above.