If I choose to downgrade a SQL Azure database from a higher SKU to a lower one (P6 -> P2 for example), does the database incur in a downtime? if so, how long?

Thanks in advance. 

From my expirience, there is no downtime when changing tiers. I've done it few times going up and down both. Change is not immediate, it takes 5-15 minutes, depends on how many tiers you're jumping and size of database. But while changing tiers my databases where operational.

Check this article for more details about edition upgrade - https://msdn.microsoft.com/en-us/library/azure/dn369872.aspx