What is the dollar value of a mile?

That will depend on your valuation of a mile.

The airlines probably assign a value of 1 cent to each mile, or at least, KF does. (1.02 cents)

But we have to look at two things: The price you pay to obtain the mile (cpm) and the value you get when you use it.

So for cpm, if you are paying money to get miles (e.g. by paying annual fees), I would set it at 1.8 cpm. So if there is an opportunity to buy at lower than that, I'll spring for it, because I know I can get it cheaper than 1.8 cpm

When redeeming, then I will be looking at how much value I get out of it, which is pretty much taking the revenue cost of a flight (sans taxes, etc) and dividing it by the miles I spent to redeem that flight. So I would be aiming for at least 3.5-4 cpm when spending it, more if possible.

Certain FFPs have sweet spots, whereby your redemption for a similar flight route will be worth more. Certain FFP will also sell miles at a discount, making it a good programme to buy and burn (but not to credit to). Kelly has listed some of them down so I won't be listing them here.

Depends on the program and this is part of the fun of the 'miles hacking' of finding the best deal.

The lower bound value is 1% (100 miles = 1 dollar) since for Krisflyer they actually let you purchase revenue tickets (-) with a cost reduction based on that.

The upper bound value is perhaps 4-8%, with lower end if the redemption is for travel on C/J booking classes (-). The higher end would be value on F/R booking classes (-). Additionally there is lots of scope of optimising based on different scheme (Avios / Alaska come to mind (-)) and promos that happen on adhoc basis (eg buy miles at special discounts like 1 for 1; essentially doubling value).

If you make the scheme work for you and you like to spend time on it personally, I'd say at least say worth 4% and feel anything extra is really nice on top but if you are busy (with cough cough real work) then use 1-2% instead.

Hope that helps

(-) all specialised terms in the mile hacking space. IF you dont' know what it means, use the lower bound of value; if you curious to optimize further (and willing to spend the time) use the upper bound :)