Can I split a 200v range circuit into two 110v circuits?
You cannot do that because of the following primary reason. The large gauge wire that feeds to that point is most certainly fed from a breaker in the breaker panel with a rating of 30A. If a short or failure occurred in your added wiring, added outlets or any thing plugged into those outlets there could be current flow well in excess of what the small 14AWG wire can safely carry. The wire would overheat, melt insulation and combust any adjacent materials. The result would be a serious fire, occupant safety concern and expensive property damage.

The proper way to do something like this is to run the appropriate sized wire back to the power panel and install the suitably sized breaker (15A for 14AWG or 20A for 12AWG). 

You could possibly replace the existing 40A breaker with a 15A breaker so that the large gauge wire is current limited to 15A but before considering to do that check with your local code enforcement agency to see if that is allowed. As a bare minimum such replacement should always label the large wires at each end to indicate that it feeds a circuit with 14AWG wire. 

There are a couple of other things to consider here too:


Check the microwave ratings carefully. Some units may very well require a 20A circuit with 12AWG wire.
If that existing 40A circuit was one that was installed some longer time in the past and terminated into a three prong outlet then there is unlikely to have been a safety ground wire in the cable back to the breaker box. You will not be able to connect outlets up to the old three wire power cable that does not have a safety ground as it is now required by code for any new added wiring and outlets to have a safety ground connection. This alone would require running new power cable with the safety ground back to the power panel.