Your attempted way should work - I just tried it with the latest geth and MEW. In case you have an old version of geth, upgrade it first and then set the password to something simpler (it seems you have a simple password typo). You can change the password of an existing account via geth account update [ENTER ADDRESS WITHOUT 0x HERE]. Then try again in MEW.
It is that simple! You do not need to import your private key from MetaMask into MyEtherWallet. To send ether to another destination (either yourself or another user's account), all you need is the account address. So, copy your receiving address out of MyEtherWallet, and put that in the "Recipient Address" field in MetaMask.
There are "gas and fees" and stuff, they're just on the next screen in the wizard flow of sending Ether in MetaMask. So, hit "Send", and then you'll see them. On that second screen, hitting "Accept" is the final confirmation that sends the transaction to be mined. You don't need anything in the "transaction data" field because you're not activating a smart contract, you're just sending Ether value. Set your gas price to 2 GWei (unless you're really wanting it to confirm really quickly), and you should be set to go!
Best Answer
It's unclear exactly what kind of advice you are looking for here. Do You need to do this personally, or are you writing software in C# to do this for others?
If you are doing it personally, MEW should have provided you with a mnemonic or private key that you can import into metamask. Once you've done that, simply copy your coinbase address into the metamask transaction UI and sending should be straightforward.
Otherwise, this should get you going in the right direction. (I havent used nethereum.)
1) get a web3 provider using infura or a local node like this or this.
2) ask the user for their intended destination address (don't use the coinbase api)
3) use web3.eth.sendTransaction to send funds using their desired signer. (MetaMask or a local node. This doesn't matter to you.)