1. VB and VB.NET are very similar. They have their roots in BASIC, a programming language used on nearly every architecture since the language's inception. Even the classic Commodores used it. Of course, VisualBasic is a long way from the Beginners All-purpose Symbolic Instruction Code used on the Commodores and the Apple ][. Over the years, things were added to the language to allow it to perform different additional functions. Before Windows, Microsoft released a version of BASIC with MS-DOS called QBASIC. This was the direct predecessor of Visual Basic, introduced to make programming Windows applications simpler, much as QBASIC was introduced to simplify DOS programming. VB quickly grew and matured, expanding functionality with every edition until it became equally as powerful as nearly any other language available. VB.NET took VB and hooked it even further into the rest of the system by connecting the language directly into the brand-new .NET framework. The only real difference between the two is built-in .NET support, though VB.NET does streamline the way some things are done.
2. I can't recommend any eBooks in particular, given that I didn't have Internet access while I was learning VB, and I haven't quite gotten into .NET programming yet (my focus shifted to web programming somewhat). However, if you can find an eBook version of the latest Sam's Teach Yourself Visual Basic [insert VB version here], you should be set with the basics, and have enough understanding to figure out most issues. Anything more complex is likely to be found only in massive reference volumes, requiring painstaking search and research. I'd ask those questions here. :)
3. VB.NET and ASP.NET (and indeed, VB and ASP) are unrelated applications, and indeed are targeted towards two very different platforms. VB is designed for desktop software development, while ASP is designed for web software development. ASP allows the use of VBScript to process information and generate web pages on the server side, as opposed to processing the same code on the client side. This makes interactive web pages even simpler to create as you only have to learn two languages - VBScript and (X)HTML. Other approaches that provide similar functionality to ASP are PHP and various CGI applications. Options that provide functionality parallel to VB are C/C++, Java, and numerous others. VB and C/C++ are the most common on Windows, both providing native code, while Java requires the use of a virtual machine. VB doesn't work outside of Windows programming, so C/C++ and Java rule supreme in other systems, as well as for applications designed to run on multiple different systems. Of course, most of this is unrelated information, but the short answer is no, you don't have to ever learn ASP.NET to learn VB.NET - or vice versa.
4. The best IDE for VB - of any version - is the one that comes with the compiler. Indeed, the Microsoft IDE is considered to be part of Visual Basic anyway, so using any other IDE wouldn't quite make sense. Since VB is a language hybrid of WYSIWYG and hand-coding, attempting to create an application using, say, Notepad would be an exercise in frustration. One of the strengths of VB is the speed of development. Instead of writing code to create and display a window (called a "form" in VB parlance) from scratch, as in C/C++, you draw your forms in the IDE and then add code to the controls. This separation of interface design from the code behind it speeds development considerably. It also gives the bonus of using controls from Windows' own GUI DLLs, so VB applications have the same look and feel as the rest of the system. Very rarely will any developer need to alter the default behavior of a built-in control.
5. Since VB.NET is a natural evolution of VB - and indeed its replacement - learning VB cannot hurt in figuring out VB.NET. However, it is certainly not required, and some differences between VB and its upgrade may actually make things more confusing. For example, VB doesn't - by default - have access to the .NET framework, nor are any of the framework's capabilities and features built in. Additionally, VB.NET introduces some coding practices that are forbidden by VB (VB doesn't support termination of statements using semi-colons ( ; ), while VB.NET does - indeed, if I understand correctly, it requires it). Regardless, if you've managed to learn a few other similar languages, and can manage to tell them apart in your head, then you shouldn't have any problem learning both languages simultaneously. Ultimately, the order is up to you.
6. It's always a good idea to use the latest version of the .NET Framework, if only because it has the most options and support. It also has the greatest degree of security and stability - much like a Windows Service Pack. There are, of course, minor advantages to using older frameworks. Computers which don't have Internet connections or otherwise don't use Windows Update might not have the latest framework installed, leaving their computer unable to run your code. However, this problem is largely mitigated by the fact that the .NET Framework is free - you can include copies of it with your applications to ensure that every one of your customers or clients will have the ability to run your software. After all, if they can't install an updated framework, they probably also can't install your application.
VB/VB.NET is a good place to look for quick and easy development of Windows desktop applications. VB is best for applications predating Win2K (since it was released in 1998), but if you're writing code for Win2K, WinXP, or Vista, VB.NET is a better bet. If you feel you don't have enough control over how your application works, you can look around online and find near-endless resources to guide you along the paths to making your application do exactly what you want - and often the Perl mantra "There's more than one way to do it" holds true in VB as well. As you become more familiar with the language, you'll become more comfortable with its limits, as well as how to ignore them. So congratulations on deciding to enter this crazy world, and welcome!