Hi John,
I've always done bottle-conditioning the ten years I've made cider (and for beer as well, though I know that's not strictly relevant here), so:
1. I use corn sugar/dextrose if I've got it, as it's neutral, fully fermentable and has little to no effect on flavor. White table sugar can work in a pinch.
2. The amount of sugar to add depends on the carbonation level you're going for in the cider (and, crucially, what carbonation level your bottles can tolerate). There are certainly more scientific ways to do it, but I use the dosages recommended for a given # of atms of pressure from the calculator site(I use Brewer's Friend, though I'd be interested to hear what others use). That site tells me a given value of sugar in oz, which I then weigh out if I'm feeling finicky or convert to volume using a powdered sugar calculator (which converts weight to volume). My target level of carb varies depending on the batch's tannin's etc., but I usually aim for between 2.25-2.75atms, which is within my margin for error for my bottles even if they had a little RS and lots of dissolved CO2.
3. After bottling, I'd bottle condition for at least a month, ideally longer. You get a lot of your CO2 produced before then, but it takes a while to equalize throughout the liquid (in my experience, at least). You've also roused the yeast, so waiting longer lets that yeast die, drop out of suspension, and hopefully form a heavier cake that will leave you a clearer product when you open the bottle.
4. I've always found it easiest to draw off enough of the cider form the batch that it dissolves my priming sugar fully (I also add my bottling dose of sulfite to this mix). Then, I slowly stir it into the whole batch again, trying to gently circulate up from the bottom so the dosage gets equally dispersed throughout. Dosing each bottle with sugar seems like a) imprecise, unless you're making a solution for greater accuracy, and b) a ton of unnecessary finicky work, ESPECIALLY if you're trying to be accurate.
3 (II). You will be oxygenating the cider some at priming, just as you do when you move it to your bottling fermentor. This is one of the many reasons to add a touch of sulfites at bottling, as an antioxidant. Some oxygen would also not be the absolute worst thing, given that you're having your yeast referment in a contained vessel; if it's a very reductive environment, you can get additional off flavors.
4 (II). I've never done that with the shaken-in-keg approach, so somebody else will have to weigh in on that. But I would think that doing so would knock CO2 out of solution, either lowering your final CO2 levels or (if you allow it to redissolve) meaning you'll start bottling after giving the yeast a chance to start working on the priming sugar, which I'd think would make it more likely to foam.
5. There's an argument that aging on fine lees (which is effectively what bottle conditioning is) allows the lees to work as an antioxidant, absorbing oxygen that would otherwise bond with the cider. Extended lees contact is also how you get some of the biscuity notes in sparkling wines, and (I think?) there's some recent research about the breakdown of lees increasing mouthfeel and perceived sweetness (though I can't find the study I'm thinking of at the moment). I, for one, prefer bottle conditioning. I think it lasts longer and ages better since there's continued microbial breakdown. That said, it will also probably be less STABLE (i.e., unchanging). But I think unless you're sterile filtering into your kegging, you've got a similar (albeit reduced) microbial presence that you would have if you bottle conditioned.
Hope that helps,
Matt