I can answer these in accordance with PnP rules. Implementation in the game may differ.
Sorry if this double posts having trouble with the forum so dunno if this has gone threw already
Would taking 1 level in barb work with the monk, do the unarmoured perks stack?
No, these are explicitly called out as not stacking. You use whichever is better.
Is barbarian rage useful for them?
No and yes. The extra rage damage only applies to strength based attacks, and monks attack with dexterity. However, they can benefit from the damage resistance. And possibly subclass abilities with more levels.
Do you lose anything by not getting to level 12?
A feat or ASI, 1 point of ki.
