This status thing is only a text object that is created when the EA is added to the chart, and then of course it can be modified. Again, it's just a text, I created this simple option one day, it's not a big deal.
No EA can work when the PC is down. EAs, Scripts and Indicators are controlled by MetaTrader, and when MetaTrader does not work, obviously no sub-programs (controlled by it) can work. The solution is to have a VPS server where to install MetaTrader and run it from there.