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Are expert advisors active regardless of while profile they are running on? Or do they have to be placed on a chart that in the profile that you have currently active? I notice when I come back to a profile that I have EAs running, they activate again. Or is that in addition to the "once per bar" activation.

EA's activate (if you have completed bars checked) like this:

every time you attach them to a chart
every time a new bar starts
every time the profile which has the chart in which the EA is attached to, is activated
every time you boot the platform

Is this right?

Instead of activate, I will say restart (restart = start from 0). In mt4 and ea will start working if:
every time you attach them to a chart
every time you change the chart timeframe

Expert advisors will continue execution if:
every time a new bar starts - it continues execution. If you trade ticks it will continue execution every ticks. They do not restart
every time the profile which has the chart in which the EA is attached to, is activated - Not sure about this one. You should not leave attached eas to the chart - especially on real account. To be safe, I always attach the ea again and remove it at the end of my trading
every time you boot the platform - same as above.