Questions:

1) What do the stars mean by the enemy name?
2). Are blue names stronger than green names?
3). How can you tell what stars or color name you pet is after you catch it?

Okay, so I see now that one star is weak, two is standard. So I was right in thinking the stars meant they were stronger and the color is rarity.

So how do I tell how many stars my own pets have and what rarity they are?

see this post which breaks it down, but
white = common = 0 trait points,
green = uncommon = 1tp (one carry)
blue = rare = 2tp (one vis, or two carry)
purple = epic = 4tp,
yellow = legendary = 6tp,
red = mythical = 8tp

star number's a bit more complicated since it depends what region you're in, but overall it amounts to how much mana the wild sylesti will have when caught (starting ay 60 mana for a 1 star and increasing by 20 for each additional, capping at 120 at 4 stars) (it presumably also affects their stats somehow while you're in a battle against them in regions 1+2 but not after they're caught, it's too early for me to go and do all the 'hunt for some different star count sylesti in r1 or 2 right now. Might do one of the dungeons later and check)(in region 3 and festival zones all sylesti have 2 stars)(in LG it gets more complicated, just see the linked page if you want to get more in-depth there)

(there is also this post in the guide that covers traps in a bit more detail)

Rarity names are only used to refer to sylesti that are caught wild/found as egglings while exploring, generally other sylesti (offspring, generated, exclusive) are referred to by their vis/carry count (eg, "it's a 2v2c")

For non-sylesti enemies star number denotes difficulty