This information is included in my Guides now but I thought it might be useful to add a post about Monster Behaviour in the 2nd Generation.
First of all, your Tame Monsters from the 1st Generation are inherited with their existing Heart Levels by your Child and, as 9 years has passed between the end of the 1st Generation and the start of the 2nd, they have become far more skillful in the interim. When you begin the 2nd Generation game, any Tame Monsters who work as 'fieldhands' will be able to perform efficiently and two Monsters from the 1st Generation set to the task of watering your Crops can cover the entire field each day.
As far as Monster Companions in the Caves are concerned, there are advantages and disadvantages to their presence. If you are trying to tame a Monster, you must take care that your Ally doesn't defeat the Monster you almost have succeeded in taming! Instead, lead him/her to the Generator or to the other variety of Monster on the screen, then send him/her home quickly before the attack on your own taming target occurs.
Spells used in the Caves/Dungeons work both on you and on your Monster Ally. If your Monster is poisoned, therefore, you can use Medication to cure him/her. The same is true with respect to Cure and Escape. When you use Escape, it is your character who actually is affected by the Spell, I believe. Your Monster will arrive next to you simply because he/she follows you everywhere.
When you are in hostile territory, therefore, you need to be aware of your Monster's status at every moment if you want to keep him/her with you. When HP reaches zero, the Monster Ally simply will return to the Barn, but that may be inconvenient if you were hoping to use his/her aid again in the same visit to the Caves.
Finally, when an Altar/Stone Guardian requires the defeat of a specific number of Monsters as its 'price', any Monsters defeated by a Monster ally appear to count towards the total. I need to test this again, though, because I was fighting at the Ally's side and may have delivered some of the blows myself...