#2
Yes, thicker strings can have a super bright sound at high tensions, Bigger strings are designed to support lower tunings, such as Drop A, B, and C, to D, to make really low tunes and vibrations, while smaller strings are designed for high tunings such as regular tuning and are easier to play at any tension, just can't tune them low or they won't make any sound other than a flimsy rubber band
#3
In principle yes - thicker strings have more mass so will be louder and sustain better. They'll also produce more harmonics so will give a "richer" sound than thinner strings. In practice though those differences are very subtle - a lot of people will struggle to quantify the difference even if you AB the two.
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#4
stings make a huge difference. Thickness doesn't make too much but the difference in different brands is huge.