1.)What are the specialties of pet and their functions and the best recommended pet for characters. why?
Go to Boost (where you "enhance" pets) then click Pet Guide. Shows you all pets & their inherent talent.
>If you have specific skill in mind (ie poison) you can also browse Pet Skills & decide which category of pets you want (Plant, Elf, etc). Skill books from market (or from skill boxes that you can pick up, but those might give you other books instead).
I can't answer for the other classes but I'm a priest and I love my mushroom king (it has poison that lasts a couple rounds & targets multiple) and it really makes a difference because it helps to end battles early & before my MP runs out (this is esp. an issue for me against bosses or in hero temple).