Tiffany uses their own grading system, which I personally (nor do most in the industry) do not find reputable. Their grading sets their pricing, and if you're watching the hen house, where's the oversight?
As far as which lab is better? GIA and AGS are about on par, they excel at different things. I prefer AGS because they properly grade cut. They're good with color and clarity but GIA is better for color and clarity, IMO, as far as they're the industry standard there, but they suck SUCK suck at cut grading, so if you're looking at a GIA report that says "excellent" cut, take that with a grain of salt because it means jack squat. They've been trying for years to set something decent up, but they have quite a ways to go. So consider GIA and AGS. Depending which vendor, most of the ones I sent you deal with AGS. They're strict on cut quality.
Ideal table, depth and angles for a round are pretty set. But there are no set numbers for any other shape as you can't do those based on numbers. So if you want a round, I can help you with that based on numbers. I have that info posted several times throughout the forum, it is in the Diamond Reference thread here
Diamond FAQ/Reference thread
I will be kind of off the radar for the next few days, so I won't be much help in this thread after today, but if you PM me I can assist you with more questions.
From a photo I could give you a guesstimate of carat size only based on a guesstimate of the width of the shank. Cut quality I can't really tell from a photo of a ring, I need much more information than that. I could tell you the shape (ie if it's a cushion, etc.) from that though.