Depends in part on the scale. Without the magnetosphere blocking the solar wind the atmosphere would have long since been blown away by said solar wind, and we would have no atmosphere, hence no weather.

Even on a smaller scale, "space weather" affects Earth's weather, and again, that effect is mediated by the magnetosphere.

Here's a link. "Simulations with a whole-atmosphere chemistry-climate model, extending from the surface up to 500 km altitude, demonstrate that magnetic field changes even affect the climate in the middle atmosphere (15-100 km altitude), and regionally also in the troposphere, near the surface."

Yes, although the ways that it happens are complex.

Earth's magnetic field deflects charged particles from the sun into striking the Earth's poles. This is a source of energy in the upper atmosphere, resulting in heating. This has yet more complicated implications, more than I can explain.