Bureau of Occult Control

"Access" occurs if a card would leave the collection or cemetery zone, either ))moving(( to another zone (i.e. the realm) or being banished. You may still freely look at your collection and cemetery, and add cards to them.

The (2) is paid when the ability which would "access" cards is put on the storyline, allowing the entire effect access. The Bureau does not increase casting costs; it is a separate cost. It will not be reduced by effects that reduce casting cost (e.g. Mix Aer), since it's not a casting cost and often there are effects that access without casting anything (e.g. Adept Illusionist, Imposter, etc.)

For example, Archimago must pay (2) to activate his ability, even though 3 cards are banished. Similarly, Deathspeaker must pay (2) to use his ability, even if on Death's Door, as this is not an increase to the cost of the ability but a separate cost which must be paid in advance.

You may choose not to pay the (2) and resolve the rest of the effect but be unable to access cards from a cemetery or collection. For example, Feast for Crows may search the other zones and simply not remove any spells from cemetery. Adept Illusionist may search your spellbook and hand for another copy of itself, but not your cemetery.

No. You must pay the (2) before even casting the spell, as a separate cost. Once you actually cast it, you resolve the effect normally and your access tax has already been paid.

You still have to pay the Bureau tax! The cards themselves are not being affected; a separate cost is being added to allow you to access them.