All of the non-GCD abilities will fire off on the first press.
At least the first press where all the conditionals are true.

The first ability on the GCD will hang or cast.
And then the next.

If you are short on characters, /use and /cast, function the same.

You could go with:

/use [nopet,mod] Summer Water Elemental
/use [pet] Waterbolt
/use Frostbolt
/use [combat] Icy Veins
/use Focus Magic
/use Presence of Mind
/use 13 [combat]
/use 14 [combat]

The macro with a mod key will summon the pet.
Without the mod key, it will not try to summon the pet, which means it cannot hang, if there is no pet and the pet is on cooldown.
I believe Presence of Mind only fires off on offensive spells, not the pet summons.
Similarly, you don't mind if a party member gets Focus magic buff early.
And you might want to not already be in combat for these to fire off.
Then again, you might want them to not fire off until the 2nd spell.