No they should cover it with a tarp till clear weather
Normally no. A wet roof is way too slippery to work on.
I doubt it .. A wet roof surface would increase risk for injuries and falls .. And .. If the roof is tarped .. By removing the tarp it would allow the roof to leak. Their tools would also be compromised during rain .. electrical tools and compressors etc can't be used in the rain.
You don't want water trapped under your roofing material. It will lead to mold and rot. They need it to be dry to finish, and to protect the warranty.
Probably not, and it's not a good idea too. If the roof is not entirely covered and sealed, working under the rain will cause it to leak into the house and start doing the damage it's supposed to prevent..
They should cover it and wait for clear weather.