I think this may be your misunderstanding. "Connect automatically" will connect to any available internet connection, whether or not any program wants to use it at that time.
It does mean that you are always online, so long as you can physically receive wifi or cellular signals & have suitable connections configured.

You say you have "Connect automatically" set. This should be enough to get a wifi symbol or cellular data symbol in your status area (i.e. connect you to the internet). If you don't have either then you have a connectivity problem not an email client problem.

If you are connected to wifi or cellular data and your email still isn't refreshing, then yes there is an email client problem.

Once again, the email client should not initiate a connection in the background. This is by design.
If you choose "send & receive" manually, then it is allowed to ask you for a connection. Otherwise it is only allowed to use a pre-established connection - not to connect, send/receive & disconnect.