Do management operations such as database copy affect our current db performance?
After executing copy database operation and database export operations, this is the data that I gathered. I used:
select * from sys.dm_db_resource_stats to real time query performance information.

I used a completely isolated Premium P6 (old P3) Database.

During the create database as copy, the source database was not affected at all. That means constant 0%cpu, 0%data_io, 0%log_write. At least in premium P6.

During the database export, %cpu and %data_io oscilated both from 2% to 5%. This means export operation effectively has some performance impact (very low, at least on a premium P6).