The Pigale area is a good alternative for a night out. Beside the Moulin Rouge and other cabaret you’ll find a bunch of strip clubs and other dodgy places that I would definitively not recommend. But once passed the strangeness of that red light stuff, you’ll find some good places.
First, the best capirinias I ever tasted are in Marlusse Et Lapin, very nice & cosy hidden place.
On the other side of the avenue Boulevard de Clichy, there are quite some options in terms of bars. The good ones are easy to spot, tens of French people are having a smoke outside. Rue Jean-Baptiste Pigalle / Pierre Fontaine are a good start.
Two very good clubs are nearby: Le Bus Palladium for Rock-mainstream electro features live music until midnight before turning into a two stories dance club. Very trendy people dressed in an typical Parisian fashion. No need for fancy shoes or suit to enter, but girls after 11pm for sure.
Le Rouge: some of the best electro/ house music, and the cool people who know how to appreciate it. Trendy and quite high class. But slightly easier to enter than the Bus.