I post here the answer I get directly from the MEW team.
Thanks for being patient and for reaching out to us.
Here are the answers to your questions:
I created a wallet using Metamask using the usual 12 words.
Metamask provided me with an ethernet address.
I bought some ETH through Coinbase, then transferred those ETH to the address defined at step 2, let's call it "my address".
All OK up to know, the correct ETH amount shows up in Metamask.
MetaMask and MyEtherWallet (MEW) are both interfaces that allow users to interact with an Ethereum address (also called an account or public address). Either can be used to generate and manage a public address. Addresses are not “tied” to one or the other service. When you generate a new address on MEW, you can import this into MetaMask in the settings section of the interface. Vice versa, when you generate an address on MetaMask and receive your 12-word mnemonic phrase, you can use the MetaMask/Mist option on MEW and interact with that address.
Neither MEW nor MetaMask stores your assets or personal data. These services are not banks or exchanges – they are simply two different doorways to the blockchain, that can both take the same 'key'. All your funds are on the blockchain itself. Passwords and private keys are in the users’ hands and are only the users’ responsibility.
Then I installed MEW on my mobile because I would like to check the balance of "my address".
Sadly, I'm not able to add "my address" to MEW. It seems to me MEW is stuck with the single address it created during set up.
Your MEW wallet and the MEWconnect app are two completely separate wallets. You cannot import your MEW interface generated wallet into your MEWconnect wallet. MEWConnect functions like a hardware wallet and the private keys to that wallet are encrypted into your device, that is why it is vital to back up your MEWConnect wallet with the recovery phrases.
So it's an entirely new, and separate wallet from your MEW interface generated wallet.
You don't HAVE to move your tokens from your MEW (browser) wallet to your MEWConnect wallet, if you are comfortable with leaving them in there then you are free to do so. Note that at the moment MEWconnect currently only supports the default tokens that are hosted on our website.
Custom tokens cannot be added at this time. If you would like to interact with a specific token, you will need to add it as a default token on our interface.
This can be achieved by following the steps found in this article
Make sure that with the MEWconnect app, that it's been updated to its most recent version, iOS - 1.0.9, and on Android - 1.0.2.
When interacting with MEW's interface (Vintage or version 5), make sure you are using Chrome.
Here is more info about MEWconnect: https://mewconnect.myetherwallet.com/#
Here are the tutorial videos for MEWconnect: https://mewconnect.myetherwallet.com/#/tutorial-videos
To view your wallet's transaction history, tokens, and any other info you want to know, you can use either;
Both are Ethereum Blockchain explorers where you can view any information about a public wallet address, token info, block #, transaction hash, etc..
All you have to do is cut, copy, and paste your public wallet address, transaction hash, token name, etc.. in the search navigation bar, and any correlating information that pertains to the address, tx id, or token info - will populate.
Let us know if you have any other questions!
BTW - Heidi from Crypto Tips is a good start to get information: https://www.youtube.com/watch?v=CU_PGNiGXE8