Yes, the Premier League should play games in other countries because it would be both beneficial to them as well as the sport of soccer. The Premier League has some amazing teams that are never fully tested because the pool in which they play in is such a small sample for the world of soccer fans. The world has many great teams and teams in the Premier League could really reach a higher echelon by playing foreign teams.
Premier League is loved by people all around the world. An enthusiast would love to have an opportunity to watch a Premier League game at a facility that they have access to so that they can experience the game first hand. People will be willing to travel to these other countries to see Premier League games and it can stimulate the economy and provide more exposure for the teams. It would also provide more countries with an opportunity to participate in the league.
I know that this is probably not the question that is relevant here, but the league should not play in other countries permanently, but would be good as exhibitions. Playing in exhibitions would definitely be important in order to promote themselves and to promote soccer as a sport around the world.
No, I do not believe that the Premier League should play games in other countries. There's a reason that the Premier League is properly called the English Premier League, it's because they play their games in the United Kingdom. It is not and should not ever become an international league.
I do not believe the Premier League should play games in other countries. I believe it is perfectly fine to set up a league that is created solely for one country, this is quite common in American sports. Leagues are often formed with preset ideals as to how the league will function and there's no reason to change those ideals afterwards.