Wikipedia tells me that SAPI 5.3 was included in the Windows SDK, so go download that. SAPI 5.3 looks like it was included as part of Windows Vista, so I don't think you can use it in any other version of windows. The library is just a COM library, which means it's not dependent on your .NET version - i.e. you could use it in .NET version 1.0 even. In fact you could use it in almost any language because a lot of them support COM in some way these days.
I'll get your easy question out of the way, no you can't use the Speech API in any way from ASP.NET on the client's computer. If you're really serious about it though, you could look into either creating a Silverlight application which uses the Speech API on the client's computer, or alternatively look into constructing the voice data on the server and streaming it down to the user. However, both options would probably be quite difficult at the moment I think.
I learnt to use SAPI 5.1 from the tutorials on MSDN, just like the link you posted. They are old, I know, but they're all still relevant. SAPI 5.x hasn't changed for almost a decade now. Visual Studio 2008 can still do all the things Visual Studio 2003 could do (well, mostly I presume) and it definitely can use a COM library, so it's possible, you just got to use your brain a bit.