According to the chayei adam (klal 131:1-6) a person might be able count omer with a bracha after sunset. Accordingly, can a person make kiddush or daven maariv on shuvous night after sunset since its a complete counting?
Sefiras Ha-Omer after bein ha-shmashos is a matter of doubt, depending on whether the count of the Omer is derabanan or de’oraisa.
If the count of the Omer is derabanan, and it is permitted to count in bein ha-shmashos with a berachah, it stands to reason that it is permitted to make kiddush and daven at this time, because the entire concept of temimos, which is the reason why poskim write kiddush should not be made early on Shavuos, draws from the mitzvah of counting the Omer.
However, because of the possibility that counting the Omer is a Torah mitzvah, one must wait until nightfall until counting with a berachah, and for the same reason one should wait until nightfall before making kiddush and davening.
Please see here for another post on the question of early kiddush on Shavuos.