I have some eggs that I picked up while on a quest.how do i make them hatch?
I need to defeat some enemys(3),but i keep losing and killing my pet.how do i win??

Egglings have a 72 hour timer that you can see in their Genetics section. Wild-found eggs can be at just about any stage on that bay, while bred sylesti eggs always start at 0/72.
Once they hit 72/72, there will be a popup when you visit their page so you can choose to help them hatch (HP boost), let them hatch on their own (Str, Int, Dex, or Agi boost), or use a philter on them.

Egglings get an extra hour added to their age count for every five nurtures they are given, while hatchlings get +1 to their lowest stat (in Str > Int > Dex >Agi order when they're even) for every five (and views to their page boost their maturity count very slightly)

Not sure where you are in the plot progression, but if you have Esior's Haven unlocked you can go to the Sylesti Plaza there to get a second party slot (the third slot is unlockable once you get to the Sanctuary of Saerielle.) Getting the party slots is basically an unspoken expectancy in Sylestia progression- there are encounters later specifically designed with the thought that you'll have multiple party members in mind.

You can also increase your Mission Limit there (missions being handy if you need to make gold, get equipment/healing items, and get more EXP on the pet you used for it)
You Equipment levels on Black Pearl are okay, you should always go for higher rarity items (mystical red > legendary yellow > epic purple > rare blue > uncommon green), and try to keep your equips close you the level of the pet that has them.
Also make sure you're giving the pets the right equipment- if they're boosting the wrong attacking stat, that's kind of pointless (in Black Pearl's case, you'll want to swap that Illusionist bracelet for something else, as Illusionist equips boost Int, while he's a physical element)