L-2 spouse of an L-1 visa holder can now obtain a general Employment Authorization. This employment authorization must be applied for separately, though. The L-2 child is not permitted to work.
Yes. A foreign national in the US on an L-1 visa is limited to working in L-1 status for a qualifying member of the multinational group (parent, subsidiary, sister, branch, or affiliated company) that sponsored him or her. If the individual entered the US on a regular L-1 visa the USCIS must be notified of any change in employment. If the individual entered the US on a Blanket L-1 they may move between any of the group members listed on the Blanket L without further notification to the USCIS. Any employment outside of a qualifying member of the multinational group will require a different type of visa.
You have a short grace period in which you may look for a new job, and have it approved by the USCIS. Barring that, you need to return to your home country.