When does the bonus from a staff of power apply to the spell attack roll or saving throw in a Wizard/Druid multiclass?

Question

A snobbish elf Wizard/Druid walks into a tavern with a shield and staff of power.

Case 1:

He casts lightning bolt (VSM) at a drunkard drinking cheap domestic ale.

Case 2:

Then he casts a profanity laced thorn whip (VSM) at the goblin handing out the cheap stuff.

Does it apply for a Druid spell if it requires a Wizard to attune? (DM has ruled it qualifies as a Druidic focus.)

Case 3:

The bartender quickly hands him a bottle of foreign wine.

But the observant wizard scowls noting the wine was not appropriately chilled. He calmly stows the staff in his shield hand (Mearls?) and cast ray of frost at the terrified bartender to teach him a lesson.

Should the wizard get the bonus if the DM rules he can stow it in his shield hand?

---

To which of these spells/cases would this section of the Staff of Power apply?

While holding it, you gain a +2 bonus to Armor Class, saving throws, and spell attack rolls.

Answers should especially address the case where the staff is temporarily stowed in the shield hand.

Answer 1

The bonus applies to ray of frost and thorn whip

While holding it, you gain a +2 bonus to...spell attack rolls.

The above bonus would apply to any spell that makes a spell attack. This includes ray of frost ("make a ranged spell attack") and thorn whip ("make a melee spell attack") because both spells specifically say that they are making an attack.

Lightning bolt does not make a spell attack since its damage is a result of a saving throw (and not an attack roll). Thus it does not get the bonus from the staff.

Supplemental questions

This bonus applies any time you are holding the staff. Though PHB rules seem to consider that the hand holding the shield is occupied and not capable of holding another object according to the rules, if your DM has allowed it then it would probably work. Given that this is a DM houserule (and thus outside the rules) though you should ask the DM how they want it to work.

It does not matter whose spell list the spells come from. The staff's description does not anywhere limit the bonus to spell's to any class.

Ability Saves vs Spell Save DC

Note that the staff does not increase your spell save DC. When an item says that it gives a bonus to ability saves, it means to the person that is using the item when they are making a save against another effect. If an item (eg Rod of the Pact Keeper) says something like "bonus to the saving throw DCs of your spells" then it increases the DC your spells have against other creatures. Note though that this is a fairly rare effect.