Ok - so I just went through my retainers and put on all the spell speed I could find, and with food ended up at 686 and repeated my test. I got the exact same 394 Burning Strikes over 20 minutes.
http://i.imgur.com/s01Djob.png
Here has been my testing methodology:
1.) Verify my stats then summon Ifrit
2.) Ifrit on Obey so OGCD usage doesn't impact GCD timings.
3.) Place Ifrit at target (didn't do this first time and lost 1 attack to movement). Cast Ruin 2 (so parse timer starts with Ifrit casting instead of use of his ability)
4.) type /echo end in chat log and stare at mini parse, hit "enter" on the first tick I see over 20 mins to end the parse.
Across 3 totally different Spell Speeds (354, 592, 686) I got the EXACT same 394 Burning Strikes. Also of note, then even my 354 Spell Speed test is faster than your high speed test (basically 3.05 vs 3.1 attacks per second). That could perhaps be attributed to our differing methodologies.
Additionally, if the spell speed buff we get (it's on the player not the pet) impacted the pet - the chances of it causing 3 different 20 min parses, with differing crit rates and base spell speeds, to have the exact same number of attacks would be super slim. It's much more likely it doesn't impact it at all, and out of everything we have here the only thing seemingly off is your 354 speed test. Everything else combined between the 2 of us basically shows speed having no impact.
I also find it interesting that Garuda's 10 min parse is exactly 200 attacks, while Ifrit seems to lose 3 attacks every 5 minutes if you assumed an actual 3 sec recast time. It would make no sense for them to mathematically slightly alter Ifrit's ability - I actually think that the combined animations from it's Burning Strike and Auto-attack cause a fractional delay in his ability usage which adds up over extended timeframes.