Work rights during the waiting period depend on the conditions of the previous residence permit. If the previous permit allowed employment, work may continue until the permit is renewed or a new one is issued. If the previous permit did not allow employment, work may only begin after receiving a new permit that authorizes employment. Leaving Germany after a residence permit expires risks denial of re-entry. Travellers from countries without free-entry agreements should avoid international travel until re-entry permission is secured. A temporary stay permit (Fiktionsbescheinigung) can sometimes permit re-entry, but not all such permits allow travel beyond Germany.
Generally, if your last residence permit granted you the right to work in Germany then you can keep working until your permit is renewed or a new one is issued. But if your last residence permit did not allow you to work in Germany, then you'll need to receive a new one that permits employment before you can legally begin work.
You are technically free to leave Germany while waiting for your residence permit, but it's getting back in that may be the tricky part. Once your residence permit has expired you've lost the legal right to enter Germany, so if you were stopped at the border you could feasibly be denied entry. You would then need to wait in another country until your new residence permit is granted, and that process could be complicated if you fail to appear at the immigration office where you applied.
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