Someone on the surface of another planet with a different mass than Earth could measure it a little or a lot more or less than ours because their rulers are different lengths than we would if they're based on the same standards that ours are based on and this is because of your stipulation:
No, everyone measures the speed of light to be "c".
I thought your issue about a mathematical meltdown would only apply if I answered "yes" to your previous question but since I answered "no", I don't know what you mean by a mathematical meltdown and again, everyone measures the speed of light to be exactly the same value "c" so there's no mathematical meltdown that I can see.
Yes, he would say the sun had a different mass because all his measuring devices are different due to the difference in gravity between the Earth and Mercury.
Yes, they would disagree about everything.
You're going to have to explain to me what this mathematical meltdown is. I didn't know such a thing could be possible.
Everything measureable and observable is different:
I'm having a hard time trying to think of something that wouldn't be put on the list.