As it turns out, part of maintaining that "public support" is not taking advantage of it. The very idea of a stable monopoly on violent escalation is obviously meant to deter violence. When it instead makes violence more likely (because the monopolist is suddenly planning to abuse its monopoly) that's quite a big change and people can be expected to react accordingly.
That is correct. It really depends on the temperature of public sentiment .
Many protests get a charitable assumption in the US as a legacy of civil rights and the Vietnam anti-war movements. However, I think a lot of that good will is eroding.
Each country has different perceptions, and ultimately, each protest is different (e.g. who shot first, is it abuse, ect)