What exactly should a winger do in soccer? ?

I was a winger in school (U.K.) .  Because I was fast.  I’d leg it down the wing, with the ball.  At some point I’d have to cross it in before being hacked down.  That was my only thought.  Sometimes there would be someone in the box. More often than not, there wasn’t.  Nevertheless get the ball in the box, it’s up to your lazy bastard forwards to be on the end of it.  Track back if you’ve got the energy or the inclination.  Stay wide to spread play, create space and offer options.

Yes, I was a winger and quite quick so I was usually the first line of attack. Staying out wide is a great idea and say the other team have a corner you should drop back to about 10 yards on from the 18 yard line ready to pick up the ball if it goes into the goalkeepers gloves so he can throw it out to you. Otherwise staying out wide but pushing in slightly when the ball is on the right hand side so that you are not completely detached from the play. If the ball is going towards the goal line on the far side you can move into the penalty area to pick up any crosses which get to the far post (the near post to you as the left winger).

A winger fulfils several important functions. He can also pull defenders out of the centre by staying out wide creating more space for forwards and midfielders coming through if the defenders come out wide to pick you up. 

Being able to cross the ball with your left foot is pretty important. Fortunately I worked pretty hard at being able to use both feet and used to take left footed corners from the right hand side so they would swing in. If you are able to use both but are mainly right footed it gives you the opportunity to occasionally cut inside, that is to bring the ball more infield from out wide and possibly get a shot in on goal as long as you have a decent angle.

When I lost the cutting edge in speed I became more of a centre forward. You can work on your speed by training in 30 yard bursts but it is the explosive type acceleration and ability to cross the ball when you have a half yard of space which is one of the keys.

Here is a video which explains many of the things I have said.

https://www.youtube.com/watch?v=2tgU_ODwpuk

I used to be captain and played centre half. I wanted the wingers to stay on the wings, so defenders could pass the ball out to one of the wingers to run it up the field on the outside and then knock it into the middle again for the forwards to score. 

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Unfortunately even though you have a set formation, such as 4-3-3 (4 defenders, 3 mids, and 3 forwards) there are variances on how coaches like to see positions played and that can vary greatly based on the level of the team and the skill of the players.

IMO, Your responsibility is to add width to the attack.  If the attack has width, then the defense will have to spread out and that will create holes.  When you don't have the ball you have to be a target for a pass out wide, particularly if the ball is coming up the middle of the field.  If you are out wide that will either make you more open or if it draws a defender to cover, it may create a hole in the defense for another player to attack the goal.  If you have the ball and are out wide, then either look to pass, cross to a player in a position to score or (if there is space) dribble the ball up field. Don't try to dribble toward the goal for a shot unless there is lots of space.  It's usually a hard angle for a shot and a good defense will be able to cut you off. 

If the ball is coming up the opposite sideline for an attack then you may move more toward the center (but not too far into the middle) to be a potential target for a crossing shot from the opposite side.  You will need to constantly adjust your position based on the position of the ball and making yourself open for a pass away from your teammates. 

It sounds like the coach in #3 really wants you to play in a  4-4-2 or a 4-5-1 formation.  While all positions should support the defense and the center forward usually stays up while on defensiv, there are many times in an attack where a forward wing will be in front of the center forward.

My suggest is to do some research on the various soccer formations online and watch games and see how the wingers behave.