AGS is on a par with GIA as far as reliability and quality of the grading goes. In the case of a princess cut it has the very significant advantage of grading cut as well as colour and clarity - and this particular stone comes out of it very well indeed.
I'm not sure what your question about "the number of facets" means - this stone is cut with a relatively common style of crown and 5 pavilion mains. Nothing out of the ordinary. As per my response in your other thread, the number of facets is NOT a valid criterion to judge beauty or quality of a diamond.
Buying a diamond online is the big point here. Let me start by making some comments on your statements:
1. "Warranties require cleaning every six months". No they don't. Many stores offer warranties that are not tied to regular servicing. And most jewellery does not require servicing every six months - cleaning should be far more frequent (and can be done at home), and maintenance far less: it's a way for the retailer to get you back in store in the hope you'll buy something else.
2. "Despite the extra cost". The issue is not the extra cost; the issue is what is the benefit and the value it brings to you. Leaving aside the issue of missing one day on the six-months inspection schedule and losing all or most of the benefits (for no logical reason), what most people identify as "warranty" for loss or damage is nothing more nor less than a partial insurance policy. Since the "warranty" does not cover the more likely event of total loss of the ring (only the main stone - no ring, no warranty) or damage to the metalwork or theft and fire, most people also buy insurance. Which covers for partial loss... so you are effectively restricting yourself to a specific jeweller, paying extra for the warranty and then pay twice to have the item properly covered.
3. "A name I can trust". That is not necessarily a matter of whether the dealer trades online, whether exclusively (e.g. Blue Nile) or not (e.g. Tiffany) - Tiffany, Kay and Gordon all have internet sites where you can order and buy. There's plenty of solid, well run stores that choose to advertise their goods more widely by using the internet; the fact that they aren't located where you are has nothing to do with whether you should trust them. Use the ample return period offered by good dealers and look at the "online order" as an opportunity to see the stone, compare it to others and return it if it doesn't pass muster for the cost of a return shipment.
4. "Around for 25 years". Plenty of big names and chain stores that have been around for more than 25 years; plenty that have gone under or changed so much that they have lost the "special" feel (e.g. Mauboussin). Same for small stores. One of my favourite jewellers is a fourth-generation family owned shop. One outlet. Sons and grandchildren involved in the business. They will be there long after my children inherit my jewels and decide they don't like them and sell most or all at auction. Don't mistake large for durable.
5. "A local store". Fair point. But, at the same time, I ask you: For what? Most local jewellers will be more than happy to clean, tighten prongs and replate rings for free or nominal fees, even if the item was not bought from them. FedEx and USPS registered post are cheap, reliable and fast if you want the original seller to do something more substantial like sizing, or replacing a side stone or upgrading the main stone. And most chains don't have bench services at all locations, with many actually having one central location to which they send work - so the only thing that changes is that they charge you for shipping in the price of the repair rather than you paying for half of it directly. I won't even get into the so-so quality of much of the bench work done, but be aware that consistency of result is not a given within some chains.
In all, I appreciate the concerns, but I think there are aspects that you need to look at in more depth than the simple distinction between "online" and "not online" purchase.
One vocabulary point: that isn't a "spec sheet". It's a grading report, or "cert" (even though it doesn't certify anything).
Now the bad news: Infinity Diamonds (the cutters/manufacturers of this particular stone) work through a selected dealer network. I believe the dealers can call on each other's inventory, and some of them have physical stores. I don't think you'd be able to get hold of an Infinity Diamond outside of the dealer's network, but I would encourage you to look at a particular Infinity dealer: High Performance Diamonds - they have a store in Boise, Idaho, and a good website:
http://www.highperformancediamonds.com - at the very least read through their terms and conditions, and see if you like what you read.
Finally, if I were to choose between Kay's and Gordon's, I personally would choose neither, for a number of reasons (not least that I dislike chains, unless they are committed to customer satisfaction in the way Nordstrom or Tiffany are). At the end of the day, what matters is how you get on with the people at your local branch, since this is the variable under your control. Stock, policies, senior management don't change, and you have limited influence (if any) over them anyway.
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). It's on hold, and if he's able to get it back in, I'm having it sent to an independent appraiser to look at (said he would ship to any appraiser from pricechoice.com... closest one is 
