My team for dailies is Onyxian Whelpling (tail, heal, lift off), Anubisath Idol (smash, reflect & stun) and Mojo (lick, heal, kiss)
For Thundering, Anub goes first, reflects, then I swap to Onyxian and go to town with him.
Every other fight I lead with the whelp.
Lift off can dodge attacks if you time it well, heal also heals a ton if well timed, and tail smack with a slow pet is nasty.
Several people mentioned Anubs usefulness above.
Mojo occasionally has great things happen with his kiss, but he often is used to suck up a bad hit the other pets are on cooldown for.
Also, OP, your team wasn't all that powerful. Your Feline gets hit EXTRA hard by the spirit- with bad luck, the spirit can take it out without you being able to do a single thing. Your spider could stand to become rare- pretty much any rare spider will have the same moveset, with the exception of the crystal spider in Winterspring and other places, which can have its Bite attack replaced by a 5 round CD 1 round duration stun.
My main pets and the skills I typically use on them:
Aquatic- Rare Shore Crab- I use Surge, Whirlpool, and switch up the heal depending on my need.
Humanoid- Rare(with humanoid battle stone) Wolvar pup- Punch, Whirlwind, and I switch up Frenzyheart Brew and Snap Trap depending on my need as the trap can be a lifesaver if it goes off.
Beast- Rare Alpine Foxling kit- Bite, Howl, Leap.
Undead- Rare Vampiric Batling- Havent really needed an undead so I just lvled it for the achievement.
Flying- Rare Imperial Eagle Chick- 99% of the time I use Slicing Wind, Cyclone, and Lift-Off.
Elemental- Rare Pandaren Earth Spirit- haven't needed an elemental so just lvled him as he was my first pandaren spirit.
Magic- Common? Mana Wyrmling - Feedback,Drain Power,Mana Surge.
Dragonkin- Uncommon Wild Crimson Hatchling - Tailsweep on bosses as 90% of the time you will go last, Healing Flame, Lift-Off.
Critter- Rare Masked Tanuki- Just because it was the closest to 25 and looks cool. Lower lvls I used Red Cricket when needed, has great skills.
Mechanical- Rare Clockwork Gnome- Metal Fist, Repair, Build Turret(Unless the pet your battling has a -dmg skill then I use Launch Rocket.)
Thanks for the replies everyone! I've noted down every pet you mentioned, and i'm going to test them one by one. The Anubisath Idol particularly looks amazing from what you said.
Keep 'em coming!