Yes, if you set the film section of the render settings to transparent, then you can save the render with the alpha channel by selecting to save as a format that supports alpha, such as .png. You may need to tweak the reflection strength, distance and falloff to control the mixture of reflection and alpha because the environment will always contribute to the reflections.
Yes you can, simply plug something else into the world shader editor's background node:
with film not set to transparent:
in this case we can turn the reflection falloff right down as we don't need to worry about controlling the reflections of the hdri.
You can achieve this by disabling shadows in the lights (this will affect other objects as well):
Yes, simply turn reflection strength to zero: