I will go on and post my numbers. First a note about stat weights: when comparing different stats, I take +dmg (or +fire) to be the baseline (weight 100), which leads to 1 crit rating = weight 63 and 1 hit rating = weight 51. The certainty of these numbers are totally dependent on the certainty of the
DrDamage addon, otherwise they are correct for my character. Furthermore, I have (from experience/personal preference) set weights of 60 for mana/5 and 50 for spell haste. The rating for spell haste is
completely speculative, I have no real info about how spell haste affects DPS. I assume 1 crit per 10 seconds for the sake of critbased procs.
Thus:
TLC: 11667 value: (((1/0.9)*(750*1.2)*(0.35/3))*100
Magtheridon's Eye: 9650 value: (54+(170*(10/40)))*100
Shiffar's: 8640 value: (225*(15(10/0.2)))*100 + 30*63
Sextant: 8220 value: (190*(15/(10/0.2)))*100 + 40*63
Icon: 6883 value: (43+(155*(20/120)))*100
Quagmirran's Eye: 6367 value: 37*100 + (320*(6/(30+6)))*50
I hope this is at least slightly readable/understandable. We often have the usual (staticdmg +) dmg*(timeeffect/totaltime) calc.
For TLC I assume 1 +dmg = 0.9 actual damage, trying to find a balance between Scorch and Fireball coefficients. I'm also assuming a 35% fire critrate and 20% nature critrate.
For Mag's Eye I am assuming 1 "miss" every 40 seconds.
Shiffar's is currently
better than the Sextant, since it's effect lasts 15 seconds (tooltip says 10). For both of these I assume 20% procrate (from wowhead)
For Quag's I assume fireball spam and 10% proc (wowhead), so it will proc every 30 seconds, then last 6 (I'm assuming it can't stack or overlap, info about internal CD is unavailable).
Serpent-Coil Braid is a bit more complex so I'll take it by itself here:
(225*(15/120))*100 + 30*63 + 12*51 + ((1067*0.25)/120)*5)*60
1067 is the average mana value of the three best mana gems.
This gives an overall value for the Braid of 5981, which is unfortunately less than even The Restrained Essence of Sapphiron (6167).
A lot of numbers here

Let me be clear that my stat weights (expecially except for hit and crit) are
highly subjective and based greatly on my own character. I do think most firemages at my item level have reasonably similar relationships for +dmg/crit and +dmg/hit though.