Differential impedance of LVDS rules and tips?
1) Three layer boards are incredibly uncommon and difficult to manufacture, and will cost you a lot more. Make a four-layer board instead. That being said, asymmetrical ground plane distances is not a problem, but will affect impedance. Use the Saturn PCB Toolkit to get your required trace widths and dielectric thicknesses. That tool has an asymmetric edge coupled microstrip impedance calculator built in.

2) The exact distance between the two conductors in a pair isn't as critical as you might think, provided they are well away from other conductors and polygons that are not part of the differential pair. Design to make sure the tracks are as close together as possible and meet impedance requirements.

3) Trace lengths between separate LVDS pairs are most important when your receivers are very time-dependent (i.e. you need to make sure all of your data reaches the receiver before the clock triggers. Otherwise you could lose data). Actual length requirements depend greatly on the transmitters and receivers you're using, as well as the frequency of the transmitted signals. When it comes to the conductors within a single pair it becomes much more critical. How critical depends, once again, on the frequency of the transmitted signals and how good your receivers are at detecting transitions.

4) Without providing us with your design it is impossible for us to critique it.

5) Right The First Time by Lee Ritchey is one of the best books I have ever found for designing for high-speed signals. It has a LOT of in-depth descriptions, explanations, tips, tricks, etc that are immensely useful. I cannot recommend this book enough.

While I have done some high-speed design I am by no means an expert, so I am open to corrections and additions to this post.

Marcus is right, though, each one of these questions could be given its own post and you'll get more in-depth answers that way.