Checking-in bags to different destinations when flying with a layover

To pick up your bags before your destination is known as short checking, and not uncommon on international flights where you have a long layover or a stopover. But you may need to jump through some hoops:

• Security personnel hate short checking, because it is a potential way to separate a passenger from a (theoretically lethal) bag. You may get flagged for SSSS on account of this.
• Airlines hate short checking because it makes hidden city ticketing (or throwaway segment ticketing) easier, and may require additional intervention (e.g. manual override) when checking in.
• For the above reasons, you may find an agent may refuse to short check a bag on a short haul itinerary, although I have never had a problem on long-haul intercontinental flights.

Yes it is possible. It is at the discretion of the person issuing the tags.

For this you need to skip the automated luggage check and go to the check-in counter, so give yourself extra time at the departure airport. Just request one tag for each destination when registering your bags.

Note that in some cases you may be forced to pickup your luggage at the transit airport even if they are tagged to your destination. Again, the person at the check-in counter is the one to tell you this.