Thicker diameter is always associated with greater stiffness which increases the intonation deviation at higher frets. Thus it's alway G as the thickest treble string which has the biggest intonation problems.
Never heard something similar about higher tension
I don't think you can generalize that in this way. Stiffnes is a material property, too. So of course, given same material thicker is stiffer, but something else can be less stiff even when thicker. Especially with wound strings it all depends on the minute details of the string design (the kernel to winding thickness ratio; kernel material, winding slope and density etc etc.)
A phenomenon of stiffer strings is that the higher 'harmonics' are not in 'harmonic' proportion to the fundamental frequency, which can make them sounding interesting or just intonate badly. This beeing said, because to make the tension higher, especially with plain strings, you have to make them thicker, if you choose the same material, so automatically you make them stiffer, and so you can indeed run into intonation problems, when the stiffnes runs out of the suitable range of values.
That's why one might consider to choos higher density materials if one wants to go to higher tension strings, because they can stay slimer at higher tensions.
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