First:
Yes, you can insure anything no matter if it's certed, non certed, etc....you get an evaluation and your insurance company will accept it or not.
As far as GIA--yes, you will have to have the diamond un-mounted in order to have it graded and certified. There is no way to accurately grade a mounted stone, because inclusions may be "hiding" under prongs, you can't accurately tell the color b/c there is contact with metal, the color of which can refract through the stone, etc, etc. So this insures the best accuracy.
NOW--to your appraisal. You can speak to an appraiser and ask that they NOT give you an appraisal for "retail replacement value" but rather for what the cost would be to purchase just that diamond. I would actually recommend having this done AFTER you have had the diamond unmounted and graded by GIA if you wish to go that route. Even if it was certified "back in the day", I'd still suggest having it regraded (if you're talking about a round) since there have been recent-ish changes with the addition of a Cut Grade--this way you will more accurately be able to compare "apples to apples" in your research comparing your diamond to the cost of others in the marketplace. Then once you know exactly what you have, you can at least look online (if not also with your local jewelers, etc) and get an idea of what that diamond would cost to buy it. You could also then use that information to go to your appraiser, and show them, and ask for the cost appraisal to be somewhere in that ball park.
Lastly, to counter the worries about a jeweler switching out your diamond: Of the gazillions of jewelers and jewelry store employees in the country and the gazillions (ya know, rough numbers and all) of customers that patronize these jewelers, stuff like this happens about once in a blue moon. Jewelers have more going on than to have the time to find an approximate match to your stone in a fake and make the switch, not to mention they're not going to risk their businesses and reputations to do it. I'm sure there's an unscrupulous jerk in the mix here and there through the country, but you're really talking about a needle in a haystack. The chances that this will happen to you are pretty close to nil.
IF, however, you still want to sweat the chance of this happening, you have a couple options--
1--first and always ANYWAY, go to a reputable jeweler/jewelry store.
2--if you're going to have your diamond certified, go ahead and spring the extra $$ for a LASER INSCRIPTION. What this means is that while the diamond is at GIA being certified, it will have a teensy tiny laser etching (visible under appox 30x magnification) in the girdle with the corresponding GIA cert number and/or any inscription you choose. This means that any jeweler, or yourself, or any friend with a microscope or a really good loupe, can look at that number at any time and make sure that your diamond is your diamond. The inscriptions is so tiny that there is nothing about it that will ever interfere with the appearance of the diamond. If you CAN'T find your inscription(s) once the diamond is set, do not fret. First look to make sure the jeweler didn't accidentally set it under a prong, and if they do, you can ask them to loosen the setting and rotate it slightly so it's free to be seen.
3--Also, since the size of your diamond is over a carat, unless you opt for a half-cert/dossier, you'll have a plot made of the diamond's inclusions. This you can look at the plot, and then look at the diamond through a loupe or under a microscope and compare the type and location of the inclusions to the coded inclusions on the cert, and verify it's your diamond. No two diamonds are ever alike so the plot of the inclusions essentially acts as the diamond's "fingerprint"
I hope my long, rambling answer has helped to set your mind at ease and help you some. Good luck with all the decisions you have to make ahead of you and congrats in advance! (and here's hoping your daddy gives ya a good deal, whatever comes of the appraisal

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