1 Reportedly, Windows 10 was going to be Windows 9, until development was far enough along that compatibility testing began. It turns out that way too many programmers that should have known better didn't bother to use any of several API calls to determine what version of Windows they were running on, and instead did a string comparison inside some internal Windows file or another. If they found "Windows 9*" they decided they were running on Windows 95 or 98.

So huge numbers of programs were malfunctioning because they thought that they were running on Win 95 or 98 rather than the new Windows 9. There were so many that management decided to skip a Windows version rather than have to create an enormous number of compatibility shims. (Or let an enormous number of existing programs fail, but MS really really hates doing that, as they realize most people love their programs more than they love their OS, and if a program fails when the user installs a new OS then the user blames the OS even if it really was the program's fault for failing to abide by specs.)

2 Ok, now why do they want to give it away?

3 New CEO, new business plan.  The bigger push here, IMAO, is to *upgrade* from things as far back as XP and reduce their support headaches.
Other views:  reflects actual reality where OS piracy is common, and undercuts potential competition from linux someday figuring out how to be usable by non-programmers.  If everybody's already on Windoze, it's easier to sell them Office (which has always been the real cash cow, not Windows) and since phones and tablets will also run the identical core OS, makes cross-platform support easier as well.
Come to think of it, that last one seems like a strategic gamble--that OS sales to PCs is less relevant compared to improving market share in portable devices where all the growth is, leading to improved app store revenue.
[This is all opinion and I have no inside knowledge]

4 The ultimate objective is to get a single OS running both mobile and desktop applications, so that Windows can have an "app store" model and MS can take that easy not-much-work cut off the top of all those transactions. Windows 8 was an attempt in that direction, but it went too far into trying to push desktop users into the mobile UI, which turns out to be a massive misfeature. At this point Win8 simply isn't going to get any sales from current Win7 customers - additional sales will represent new machines and some of THOSE will get downgraded.

MS is more or less abandoning the idea that current desktop users of Win7 or Win8 are going to upgrade to Win10. There's simply not any compelling reason to do so. But at the same time, Windows' presence in the mobile market is tiny; it makes no sense to develop your mobile app for Windows phones or to port over your desktop app either.

If MS can make such porting trivially easy, and if they can get the installed base on the desktop, then things get interesting. It's still a tough market to push into. Apple has a big advantage at the high end of the market, simply because they've got the status-symbol advantage. Android is going to be the item of choice for those who don't want to pay the Apple tax. MS's only real chance is to get that unified market - consumers who have desktop or laptop machines, who can say "well, if I get the Windows then my app stuff will work across every device I have".

But that doesn't work if most of your desktop users aren't on the cross-platform version of your OS, right? And at the same time, very few people have reasons to upgrade the old boxes on their own, right? So... upgrade people for free, and get everyone on the cross-platform Win10 (assuming it doesn't suck). The revenues you're giving up were largely illusions in the first place given how few people are going to upgrade on their own. But if it lets you develop the critical mass you need to make the mobile plans work...

5 Another puzzle piece: newly-released version of visual studio is supposed to do iOS, Windows & Android apps natively from a single source set in their .NET code.  Write once, run anywhere.

6 Most of the Windows OS sales are from new computers, so the fractured Install Base costs them more than upgrading all Windows 7 users (in theory) to Windows 10.   It's actually a good idea, in general, given the way things have gone in the Desktop space.

7

I wonder if this means Microsoft is trying to port Win10 to ARM? Because ARM pretty much owns the tablet and phone market now.

If Microsoft is chained to x86, they're doomed.

8 Windows 8 works on ARM just fine. They even sell a hardware line for it, called "Microsoft Surface" (not the "Microsoft Surface Pro", which is x86 based).

9 BTW, MS sustained a curious loss in automovite area recently, where Ford killed the Windows-based Sync and rolled a homebrew (on top of QNX) for Sync 3. It's available in 2016 Fiesta and Escape, but will come to all other cars eventually. This comes right at the moment when Apple and Google are getting serious about onboard apps. MS squandered their early success here.

10 Indeed, Microsoft is so eager to push people to upgrade that they've pushed an update to Win7 that puts the infrastructure in place to eventually give people nag messages to upgrade (assuming MS chooses to leverage it).

ExtremeTech: Once more, with nagging New Windows update will 'notify' users to install Windows 10

11 Yeah, I've seen that. There's an icon in my tray which won't stay hidden when I tell it to be. And the Windows Update control panel gizmo now has a "Win 10 upgrade" link in it.

12

I'm not massively surprised that Windows can be ported to other platforms. The original Windows NT 3.1 install came with four versions for four different platforms, and I can believe that someone prudent has been trying to maintain portability all this time.

Microsoft has had a long and very profitable symbiotic relationship with Intel but they always knew it couldn't last forever, and didn't want to close the door to working with other kinds of hardware.

13 Yes, originally on Intel, PowerPC, MIPS, and Alpha.  They also supported Itanium from Windows XP through to Windows Server 2008 R2.

14 Right at the time of NT triumph over the proprietary Unix, Sun tried to get Microsoft to run Windows on SPARC (I don't remember who was the chief of SunSoft at the time, but I'm sure Scott approved those overtures). So, around 1995 we designed the endian into a reference implementation of the upcoming v9 chip that we did. Windows only worked on little-endian architectures. Possibly still does. The indianness switch adds a bit of time to some paths, but modern CPUs should be able to amortize it easily (back then we had 4-stage and 9-stage pipelines and not a whole lot of room to maneuver).

15 My parents are not technophiles, and extremely leery of putting anything on any sort of "cloud" ever since The Fappening. The fact that Windows 8 actually mirrors all files in My Documents, My Pictures, etc. to the cloud automatically for any registered user account is one of the biggest drawbacks in their eyes. It doesn't help that they're getting on in years, and can't remember any complex passwords, so their account passwords are the very definition of "insecure." 
My question is, does Windows 10 also silently put everything in registered accounts onto a "cloud?" My parents would rather lose all their tax documents and pictures than have them spread across the internet, and their vigilance seems to be paying off, if the literally daily calls they get trying to beg and intimidate them into sharing any scrap of personal information with strangers is any indication.

16 The beta release of Windows 10 has an option to install without connecting to a Microsoft account, so that everything is local.  

I don't know exactly what will be in the final release, but it's certain to have something along those lines, because Microsoft have to sell it into corporate accounts who would never install it otherwise.

17 You'd think they would have done that with Windows 8, too, but I had to create another proxy Microsoft account and swear up and down that "Ragtimer Threadbare" is totally not the same person as "Tatterdemalian Threadbare" before they would let me set up local accounts for my parents.

18 You can opt out of cloud sync and logging on every time with 8.1, but I think you still have to activate with a Microsoft account.

19 Windows 8 ran on ARM. The ill-fated "Windows RT" was the ARM port. I have an ARM-based Surface. It's not bad, just too big. It's the same physical size as the x86-based Surface Pro (which I also have). Windows Phone 8 also runs on ARM.

Windows 10 runs on ARM. There is a Internet-of-Things version for several small computers including ARM-based such like the Raspberry Pi 2. Windows Phone 10 also runs on ARM.

As for why MSFT are giving this away for free, that's what they essentially used to do with Windows 1.0 which was then available as a static library for developers who targeted Windows but sold to users who didn't have Windows. Once Windows is the established standard not only on the desktop but on the other two "screens" as well ("the three screens" is how some press release called the desktop, mobile and TV), licence fees will come in again. But currently users not upgrading wouldn't bring in money and users upgrading doesn't have to be bring in much more. It's only important that users upgrade so we can leave the world divided into lots of Windows 7 and some abandoned Windows 8 users.

Incidentally, fun game with the version numbers:

6.1 = Windows 7 (6+1)
6.2 = Windows 8 (6+2)
6.3 = Windows 8.1 (8+1=6+3)
6.4 = Windows 10 (6+4)

Windows 8.1 was Windows 9.