Imagine that you are sitting in your room in 2008, wrapping up some code in C++, which involves peer-to-peer networking, cryptography, and some new fancy verification protocols for something that you just invented with your giant brain called "proof-of-work." You've just solved one of the oldest computer science problems of all time, moderately impressive given the nascence of computer science. You've tested your code extensively and you know it works. It's time to send your child into the wilderness to see if it can survive.
You are Satoshi Nakamoto and you've just compiled the first version of Bitcoin. What parameters do you choose?
These parameters are the easiest parts of the code to manipulate, but perhaps the most challenging to chose. How many Bitcoins should you issue - since they are divisible, where should you put the decimal place? Should they be limited, or not? How quickly should they be disseminated - should it be over a year or a century? How often should transactions be bundled into chunks that allow for checkpoints, the virtual version of a clearinghouse?
He probably chose 2.1 quadrillion base units, or "satoshis" to permit total global dominance in 3 or more financial categories (store of value, currency, etc.) without the need to divide further: if 1 satoshi were worth 1 US cent, then 21 million bitcoins (the ultimate limit) would be worth 21 Trillion US dollars. So he's safe there.
He probably put the decimal place after the "1" in 100,000,000 satoshis because it puts a much more palatable number in our mouths. Now we're only dealing with 21 million bitcoins, something that anybody can chew.
He probably chose a long dissemination time to allow for regular people to get involved, and benefit from the technology. 1 week would be woefully inadequate as all bitcoins would be mined by himself and perhaps a few others. 1,000 years seems a bit... optimistic. About ~100 years should do it.
Now, what should he do about the rate? Well, if he knew much about gold, which he did, then he might find it advantageous to mimic the way in which gold has gone from easy to carve out of mountains by the tonne, to sifted in rivers by the pound, to mined by the ounce, to processed by the gram. The harder you try, the less you find. And this is exactly what he did.
Satoshi chose to begin with 50 bitcoins being "mined" roughly every 10 minutes for the first 4 years, with a subsequent halving every four years thereafter: 25, 12.5, 6.25, etc.
But did he chose rightly?
That Bitcoin is a technology is well-established. What is also well-established is that the vast majority of technologies are adopted along an S-curve. In the beginning, early users pick up the tools and begin to tinker, making the tech more useful. This draws in more users and so forth, until the market goes parabolic, and then slowly levels off as the addressable market is saturated.
My bet is that Satoshi considered an S-curve mining reward, but ditched it in favor of the logarithmic function (technically, it's pseudo-logarithmic, as a logarithm does not have an upper bound. Really, it's a simple series for n(2)=n(1)/2), just for the math whizzes out there), because he had to make a trade-off. The trade-off was between a) Rapid bootstrapping and securing of the network, and b) Disillusioning the medium-term adopters who might have enthusiastically embraced the technology but feel that they heard about it too late.
It's a tough tradeoff, for sure. On the one hand, having an initially rising mining reward would have been somewhat exciting, knowing that investing in mining equipment early might position you for thicker profits in the future. Mining would have been a mid-termers game, as opposed to one for early enthusiasts. Instead, all miners know that they have very short - and frustrating - windows in which to snatch a profit before either calling it quits, or reinvesting in more equipment, thus extending their risk.
On the other hand, poorly incentivizing mining in the beginning would have risked the project, which may have fizzled before it got a foothold. Moreover, a competitor, or rather, a clone (let's call it what it is), could have launched with a more generous mining reward function, and enticed miners to divert their resources accordingly.
The early days were clearly critical to Bitcoin's success, and the pseudo-logarithmic mining schedule can be credited with securing Bitcoin's place as king atop the crypto currency realm. That it has resulted in some very wealthy early adopters is a benign byproduct of this choice. Satoshi set the mining reward at full-tilt from the get go, sucking in the time, and imagination, of everybody who followed.
To close, I will reference and not break Betteridge's law of headlines: "Any headline which ends in a question mark can be answered by the word no."