Have a look at https://docs.makie.org/v0.19.1/examples/blocks/axis/index.html#auto-reset_behavior and the sections Controlling Axis Aspect ratios, Controlling data aspect ratios, Linking axes.
How to get the plot’s value limits?
If by plot's value limits you mean the axis limits, then try
f = Figure()
ax = Axis(f[1,1])
((xmin, xmax), (ymin, ymax)) = Makie.limits(ax.finallimits[])
How to get a right-handed 3D coordinate system?
That’s a good question. I don’t think there is currently a build-in option to do that.
There are some transformation methods, like Makie.scale!, Makie.rotate!, but they can only be applied to plot elements like Lines, ScatterLines <: Makie.MakieCore.Transformables but not to block elements like Axis, Axis3. (Also: Blindly applying it to ax.blockscene rips the whole axis apart 
).
Maybe @jules has an idea for that?