If you have imported data from your broker and not applied any specific time settings in EA studio then the time of your trades made on your demo and shown in the EA studio journal should match. Also start of course by checking using the MT4 backtester on your broker MT4 as discussed also.
As you say the time isn’t that important if no time settings are applied and especially over a large time period (It could be significant for only a weeks results but will be insignificant backtesting over a couple of years for example).
Also, I honestly just don’t think there is any need for different data for generating strategies with EA studio other than the premium data\historical tool. I heard of some people optimizing strategies for their broker data but personally I think a simple check that your strategies equity line is similar between premium and broker data over the same time period (i.e. for 6 months if that is all the broker data you have) is sensible enough.
As for the concern over the two live servers giving different results, that does seem odd. What are you comparing exactly? Do you mean that you have put identical robots on two separate IC market live accounts with different servers and the results are very different? Or are these checks within EA studio. Seems like a different issue, possibly due to using a short time frame and poor latency (Although in that case I would only expect the actual profits to be different but the times of entry an exit should mostly be close, could be a bit different for stop loss and take profit exits though of course). What latency do you have with your broker? I am assuming you are trading on a VPS?
There is a lot being discussed here actually which can always be a bit confusing. For problems like these I would always advise breaking the problem into as small pieces as possible and avoid trying to solve too much at once.
But I am still sure all will become clear to you once you work through the steps as discussed 🙂