How does different time-zones affect the EA working correctly? If the EA works within a certain time, e.g. GMT+3, but your time-zone is GMT+2, do you have too change the end and last hour?
The price is same in case of the price is moved online irrespective off timezones.For example, if I have the price on the chart as 1.2000 in my chart's timezone so you will have same in your charts/broker's timezone.
Of course, there are some EAs which are working within some hours (timefilter) and so on - and yes: in this case you should know (or EA should "know"/detect the timzone in automatic way). But it depends on the coding sorry.
It depend on how your EA's logic was coded, if it was made to be trading only on certain timezone, yes it will really affect your EA's performance.
For example, most Scalping EA and traders trade very well during the " Tokyo session ". The Trend follower do very will during the " London session " and the trouble maker misbehavior perform quite well during the " New York session ".