Most of the teams I used were inspired by Wowhead comments, thanks!
ROUND ONE - Random Rotation of Misc Trainers
First: Twilight Spider (2, 2, 1)I think any spider will work in this encounter, so I just chose one in my collection with the most power stat. Since its utilizing mainly dots and Leech Life, power is ideal for this pet. Some extra health doesn't hurt either!
Second: Terrible Turnip (2, 1, 1)
Third: Pet Bombling (1, 2, 1)
I opened with Brittle Webbing and then used Poison Spit. After that, I just rotated Leech Life with either dot, making sure to keep both up. Nairn went down quickly and easily.
Next up was my Terrible Turnip against Stormoen. I waited until Call Lightning was used, and then replaced that weather with Sunlight. After that I used Tidal Wave to knock down any turrets up and to damage Stormoen. I tried to save Leech Seed for Sunlight for increased healing.
In the end, my Turnip had enough health to soak the next pet's attacks (Dodge, Prowl and Bite) allowing my next pet an advantage over Summer.
I used my Pet Bombling to finish off the fight.
EDIT - A more reliable team
First: Kun-Lai Runt (1, 1, 2)Start with Mangle and then Thrash. After that use Deep Freeze to stun Nairn, which will cause Kiryn to swap to Stormoen next turn. Swap to Lil' Ragnaros just as she switches.
Second: Lil' Ragnaros (2, 2, 1)
Third: Darkmoon Zeppelin (1, 2, 2)
Use Flamethrower and then Magma Wave to knock down the turret. After that Conflagrate should make quick work of the rest of his health. Lil' Rag should have some health left to eat some of Nairn's health, who will get swapped back in once Stormoen is dead.
Damage Nairn as much as possible, and then use Kun-Lai Runt to finish him off once Lil' Rag is down. The runt should be able to take on Summer for a little bit. If he doesn't outright defeat the wolf, use the Darkmoon Zeppelin to finish off the fight.
First: Spawn of G'nathus (1, 1, 1)I started with Spawn of G'nathus and Swallow You Whole. Dive on the next round to avoid Au's Gold Rush attack. Swallow You Whole finished off Au nicely.
Second: Electrified Razortooth (1, 2, 1)
Third: Fel Flame (1, 1, 1)
Against Banks, I used my Electrified Razortooth. Even though Banks is a Critter, I avoided using a Beast pet since Banks' main attack is Mechanical (strong against Beasts).
I used Rip, Blood in the Water, and then Rip until Banks was about 50% health and his heal was on cooldown. At that point I used Devour which defeated Banks and healed up my Razortooth. It had enough health to place a Rip or two on Lil' B.
Lastly up was my Fel Flame. I used Immolate first and foremost and then Burn while I waited for Lil' B's Extra Plating to fall off. I probably didn't have to wait, since the Fel Flame hits hard and is strong against the mechanical pet, but I wanted to be sure that Conflagrate hit as hard as possible when I used it.
First: Crow (2, 2, 2)Started with my Crow and used Alpha Strike first. Once Carpe Diem used Cleansing Rain, I changed the weather using Call Darkness, followed by Nocturnal Strike and then Alpha Strike until the fish is down. My Crow had enough health to take on the next pet.
Second: Onyxian Whelpling (2, 1, 1)
Third: Legs (2, 1, 2)
Since Call Darkness was still up, I used Nocturnal Strike on Spirus first. My pet didn't have enough health for 2+ more rounds, so I used Alpha Strike until it died. If it had had enough health, I would have used Call Darkness again and then Alpha Strike and/or Nocturnal Strike.
Once my Crow was down, I swapped in the Onyxian Whelpling and used Tail Sweep to finish off Spirus. The Whelpling had plenty of health to face the next pet.
Against River, I used Tail Sweep and then Healing Flame. I saved Lift Off for when Whirlpool had 1 round left on it. This meant my pet did get hit by the first Dive, but thankfully my heal was enough to negate it. Tail Sweep to victory!
I had Legs as my third pet, but didn't even need him.
ROUND TWO - Celestial "Boss Pets"
I used the same teams and strats that I used in previous weeks (except for Xu-Fu, which I used a back up team for), so I'll be omitting the strategies I employed and instead just list my teams for each Celestial.
First: Dragon Kite (2, 1, 2)As I already mentioned, my team and strategy from previous weeks did work, but I made one mistake that cost me the match. I ended up having to use a back up team with the same strategy.
Second: Ancona Chicken (1, 1, 2)
Third: Menagerie Custodian (1, 1, 2)
The mistake made - Used Ion Cannon on a turn that Xu-Fu healed, which mean it didn't finish off Xu-Fu and it put my pet on cooldown. Hopefully I won't make that mistake again (but I probably will :P)!
Anyway, my back up team had the same strategy as my main team. I used my Dragon Kite first, except because it's faster than Xu-Fu, I used Lift Off first. This allowed me to avoid Moonfire damage (strong against Flying) and replace that weather with Call Lightning on the second turn.
After Call Lightning was out, I switched to my Ancona Chicken and used Flock until it was dead. RNG was kind and Xu-Fu missed one or two of its hits.
When my Chicken died, I swapped my Dragon Kite back into battle. Again, I used Lift Off first to avoid Xu-Fu's Feed (damage and self-heal), but I probably could have used Call Lightning too.
I switched to my last pet once my Dragon Kite was down. This time I did NOT use Ion Cannon right off the bat, and simply DPS'ed down Xu-Fu properly. :P
EDIT - Possible Chicken replacement
Lil' Bling (2, 1, 1)The strategy with Lil' Bling is still the same. Spam Inflation to add double damage to Xu-Fu until either Xu-Fu is dead or Lil' Bling goes down.
First: Zandalari Kneebiter (2, 2, 2)Yu'la (dragon)
Second: Zandalari Anklerender (2, 1, 2)
Third: Zandalari Footslasher (1, 1, 1)
First: Gregarious Grell (1, 2, 1)Chi-Chi (crane)
Second: Hopling (1, 2, 1)
Third: Moonkin Hatchling (1, 2, 2)
First: Sen'jin Fetish (1, 2, 2)
Second: Jade Oozeling (2, 1, 2)
Third: Darkmoon Eye (1, 1, 1)