Can I transfer ETH between apps? Let's say I have Blockchain wallet, and I want to use another andoid app, or other wallet. Can I do it? Will there be a fee?
Yes. You can send ETH to any compatible app or wallet. It will just take a small amount of ETH to pay the fee for the transfer (it's called gas). Gas price can be small, sometimes 0.004 ETH or less. It depends upon how busy the network is with transactions. 

https://etherscan.io/chart/gasprice