As I know, we don't have to mine to make transactions done in Bitcoin testnet, but in ethereum testnet, we need to mine to make them done. Are there any ways to make them done without mining?
No, you don't need to mine.
In the simplest setup, you will have a node to connect and it will broadcast your transactions and listen for new blocks.
The only exception is when you want to run a private chain all by yourself for development purposes. Someone has to be mining (you) so blocks can be discovered in order for the transactions to be included in a block.
In case it isn't clear, this requirement for "someone" to be mining is taken care of for you on the public testnets and mainnet.
Hope it helps.