Expanding a little on the last part, Earth's orbital velocity is about 29.8 km/s so that's the speed at which we would suddenly be leaving the former location of the solar system in a direction that depends on what time of year it happened. Regardless of direction though, the escape velocity of the Milky Way around where we are is about 544 km/s so there's no way we'd be leaving the galaxy. On the other hand the plane of the galaxy is only about 6 degrees off from the galactic center at the moment, so if this happened at the right time of year (don't know when that is) we could launch somewhat towards the core. We would not however get very close to it because the sun's own orbital velocity is about 230 km/s so we'd still be in close to the same galactic orbit overall, just potentially a bit more eccentric.

A bit - probably weeks to months.

no lol
It goes from 85 to 58 in 12 hours right now in reality world

"A bit" = 1 day, and by the end of that day it'd be freezing (below freezing if you live in whiteistan)