Can I send eg. 0.01 bch to exchange for testing, once no any problem then send out all of them?
Sure... why not?

Just be careful that you don't send less than any minimum deposit amount specified by the exchange you send to, as some exchanges/sites will just "eat" deposits under a minimum size (no refund etc).
Also, fees in BCH are really low due to large empty blocks, so you can generally send with minimum fee.
