So let's take this bit by bit.
Does the trainer have HP?
Yes, the amount of HP you have depends on the level of your green Canari plush.
No Green Canari Plush = 100 HP
Level 1 Green Canari Plush = 150 HP
Level 2 Green Canari Plush = 200 HP
Level 3 Green Canari Plush = 250 HP
When the red border appears on the screen, you have taken a lot of damage and will black out soon.
When you are not being targeted by an opposing Pokemon you will regain 20 HP/second.
How do you take damage?
When hit by an opposing Pokemon's move, you will take damage according to 2 factors
1. The base power of the move used: this table was copied from the Serebii Page.
| Move Base Power |
Damage Inflicted |
| 25-49 |
30 |
| 50-74 |
40 |
| 75-99 |
50 |
| 100+ |
60 |
| Rogue Mega Evolution Special |
70 |
2. The base stats of the opposing Pokemon and other factors (also called Total Power) This will create a multiplication factor of this damage from 0.5-1.5.
While Serebii does not include all the factors, this is likely why Alpha Pokemon do a boatload of damage to you.
Source: https://www.serebii.net/legendsz-a/playerhealth.shtml
I will update this as more information is available.