You need to use an impedance calculator tool to figure out the geometry for routing your differential pairs. If your schematics call for 50-ohm single-ended routing, you would want to use 100-ohm differential routing. Punch that into the tools along with your board and trace geometries and it will tell you how thick your traces should be, and how far apart to space your differential traces.
I believe the 100-ohm differential equates to 50-ohm single ended because you can think of the two 100-ohm impedances as being in parallel resulting in an effective 50-ohm single-ended equivalence. FWIW, the calculator I've used in the past is called Polar SI8000 which apparently has been superceded by Speedstack PCB.
Edit
I think you sould just remove the Balun, inductors, and capacitors if you are going chip to chip differential. It's not trivial to match differing characteristic impedances by putting in-line passive components on the transmission path. You would need to add an appropriate sized "stub resistance" at a carefully chosen location along the transmission path using engineering tools like a Smith Chart. Stuff like that is really only practically applicable over "large" distances with respect to the wavelength though. That's why I'm suggesting just routing pin to pin with differentially controlled impedance.
If you want to keep the capacitors and inductors for differential routing to optimize the power transfer, page 16 of the datasheet shows some formulas that "provide good starting values" for these components...
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.
Best Answer
No! You need to adhere to a defined distance to get a defined wave impedance. What you describe is a coupled microstrip line.
... in a defined distance to get a defined impedance.
In fact, you don't need a surrounding GND plane on the same layer – practically all field will be between the two differential conductors; what would be good would be a plane below!
So, that's a high-impedance load and not really a low-impedance source. I'd recommend having two matching networks: one at the source to match the source to the transmission line impedance, and one at the sink.
You could then use an arbitrary transmission line impedance, e.g. the microwave-typical 50 Ω or the 75 Ω. In theory, 200 Ω should work (and would save you the source matching), too, but it might be hard to build using your PCB materials – it depends, can't tell without knowing with what you're working.
They are not contradicting. A perfect transmission line does not radiate, so your "as close as possible" simply isn't right – yes, close, but not "as close as possible".
Use a specific calculator to calculate the right dimensions for a coupled microstrip line on a PCB substrate of your PCB's thickness, with your PCB's \$\varepsilon\$, and on the frequency you work on.