So, straight after joining the Companions, I finished Taking Up Arms quest and the Proving Honor quest.
I figure I can by now do misc. quests to help me get some coin, however, as I talk to any of the questgivers, they all tell me to go see Skjor, which in turn makes me become a werewolf in order to progress in the Companions.

Is there a way to not start The Silver Hand, but to get the misc quests?
My new character is level 10ish and I prefer not to become a werewolf since it prevents me from getting Well Rested bonus, and I can't be bothered doing smithing/enchanting/alchemy without Well Rested.

No. You must continue with The Silver Hand at this point.

The loss of the well-rested bonus should not make much difference; if you feel the need to make up for it while training, switch to the appropriate stone (warrior/mage/thief) when training those skills.