I think the correct answers to your questions are:
No, you keep any abilities and bonuses from your class so far.
No, as stated in the Rules-section for multiclassing "ff you already have the Unarmored Defense feature, you can't gain it again from another class." So you will only be able to use the unarmored defense feature of the barbarian class.
Also, in order to multiclass like this, you'll need a minimum of Strength, Dexterity, and Wisdom 13.
As a level 5 barbarian, the biggest bonuses you'll lose are:
1) You won't get Monk Unarmored Defense at all.
2) Barbarian Extra Attack at 5 won't stack with Monk Extra Attack at 5 - at 5 levels in both classes, you'll only be allowed to make 2 attacks per attack action.
I have a Half-orc barbarian (Zealot, worships Bahgtru) that I would like to dual class to become a monk.
As a level 5 barbarian, I have +10 to my move, and my unarmoured AC is calculated as 10+DEX+CON.
For a monk, when they hit level 2, they also get +10 to move, and from level 1, unarmoured AC is calculated as 10+DEX+WIS.
My questions:
1. Will I lose the barbarian bonuses when I change classes?
2. Can I choose which AC calculation to use, or do I have to switch to Monk rules?
3. When I hit level 2 as a monk, will the Monk move bonus stack with the Barbarian bonus or replace it?
I think the correct answers to your questions are:
So you will only be able to use the unarmored defense feature of the barbarian class.
Also, in order to multiclass like this, you'll need a minimum of Strength, Dexterity, and Wisdom 13.
As a level 5 barbarian, the biggest bonuses you'll lose are:
1) You won't get Monk Unarmored Defense at all.
2) Barbarian Extra Attack at 5 won't stack with Monk Extra Attack at 5 - at 5 levels in both classes, you'll only be allowed to make 2 attacks per attack action.