International scrimmages are played at the game time of the home team's country.
If our teams were to play at your arena, the game would be played at 18:30 CET. If we were to play at my arena, it would be at 02:00 CET (on Friday) which is 7:00 pm Thursday where I am (CST = UTC-6). If you wanted to watch your away match live, you would have to stay up late.
NT matches are also played (on Monday) according to where the game is played. If you click on NT Matches over on the left menu, you will see that on 2/11 that the first game was played in New Zealand at 07:00 UTC, and the last was in Mexico at 02:00 UTC on Friday, but all of these were in the evening local time (there is a block of games in Italy in the middle of the list, these were neutral site games played as part of the repechage).
Buzzer Beater shows times in the local time of your computer. If I look at your schedule it, it shows that Sweden plays at 11:30 am, and if I look at the TL (it is 19:05 UTC), players with ending times of 1:06 pm have a deadline in minutes.
Yes, players could be injured before they arrive at your team.
One other thing. I'm not sure how Daylight Saving Time is handled. The USA (where the servers and developers are located) switch a to DST a week or two earlier than Europe. This might have the effect of making your games be played an hour earlier until Europe switches.