Coming back to this, it may not be straightforward.
You will need to use several items working together to accomish this.
First the unlock door desire:
You could use presence (phone,fob) and the I'm back routine to trigger your home mode.
Or you could use the lock software referenced above to detect when anyone, or specific person unlocks the door to trigger your home mode.
Then upon house going into home mode the smartthings Ecobee software would see that and set the ecobee to home mode as you desire.
It's a multi item, step event to get the end result you desire. There is no direct lock to ecobee connection.
The away function you desire has an additional item you want to detect so it will complicate it slightly more.
If you didn't also want to also detect the door being locked.
You could use just the not present and your goodbye routine to start the cycle.
Or you can just use the door locked by outside keypad as a trigger to set your house to away mode using the lock software.
Again the smartthings ecobee software will see your house go into away mode and set the thermostat into away.
By wanting everyone gone and door locked. Which is kinda redundant, because when everyone is gone the door should already be locked. You'll have to possibly delve into Core smart app or something else.
In my house I have my goodbye routine lock my back door. This is only a backup event. Typically when we leave everyone is in the habit of pushing the lock button on the lock outside keypad. We currently use presence to trigger our goodbye routine. I could use the lock keypad but I went with presence in case someone forgets to lock the door when their running out the door at 7:59 am for their 8:00 am college class.
Then of course you could always use the built in capabilities of the ecobee to do this on its own.
If your ecobee can see motion in the house and you have smart home enabled. It will automatically enter your home mode when it is in the away portion of your daily program. Or vice versa enter away mode with no motion when your in the home portion of your daily program.
Soooo possibly place remote sensors by the main path of activity, or rooms you typically occupy when your home. The ecobee should see you and switch modes for you automatically. This being an ideal situation. You could then back it up or force it to switch sooner with smarthings telling it to switch.
I hope that helps, I am typing on my phone so forgive my typos and grammar ect.....