[RPG] Can the Robe of the Archmagi increase the DC and spell attack bonus of Cleric Spells

Yes, you can use War Priest but not if you use Multiattack

Taking a cue from András's answer, the attacks listed in a creature's stat block are not Actions but attack modes or melee attacks. Here's what I think you're missing:

Page 192 of the PHB defines the Attack Action:

Attack

With this action, you make one melee or ranged attack.

The PHB further defines what a melee attack is and, in page 195, it explicitly states that:

Melee Attacks

A typical monster makes a melee attack when it strikes with its claws, horns, teeth, tentacles, or other body part.

With the above reading, the PHB confirms that Claw, Gore, Bite, and Slam are melee attack options that can be chosen when a creature uses the Attack Action and therefore able to trigger War Priest.

But this is where it gets weird:

Multiattack is a distinct Action that details what attack options are used by the creature when the creature uses Multiattack (not the Attack Action!). Since it is a separate Action (like Ink Cloud and Swallow) it cannot trigger War Priest.

This means creatures using Multiattack cannot trigger War Priest; they must use the regular Attack action and then use a Bonus Action to attack with War Priest, but this obviates the need to use War Priest as the Druid/Cleric can already attack twice with Multiattack.

As pointed out by KorvinStarmast, a beast with different attack options in the Multiattack Action like the Brown Bear could potentially get a slight damage increase if the Druid chooses not to use Multiattack (Claw [2d6+4] + Bite [1d8+4]) but instead use War Priest to attack twice with Claw (4d6+8). This allows for a unique dynamic that a Druid/Cleric can use during combat, when needed.

No; you can only use slots to cast lower-level spells from your class

The PHB rules (p. 164) for multi-class characters specifically address a multi-class character with slots of a level they can't "know or prepare". And it says You can use those slots, but only to cast your lower-level spells. It further elaborates that if any of those lower-level spells are improved by being used in a higher-level slot, you may get those benefits.

If this loophole worked, a similar loophole would exist for wizards: they can only prepare spells in their spellbook. And the rule for whether they can put a spell into their spellbook is only that the spell must be 'of a level for which [they] have spell slots'. So a multiclass wizard with an up-level slot can find a higher-level spell, and then could copy it into his book, and then prepare and cast it. RAI is pretty clear this won't work (after all, RAW explicitly says it won't, it just also kind of contradicts itself by implication).

In other words, the multi-class character rule saying you can't overrides normal rules for learning and casting spells. It is the "more specific" rule. The character may not 'know' or 'cast' spells higher level than their class levels allow. Which means your example wizard can't copy misty step into their spell book; that would constitute 'knowing' the spell. Nor could they prepare and cast it.