Are you talking about the Universal Tween Engine(aurelienribon), or the built in tween engine within libgdx(Interpolation)?
Either way I personally haven't come across a use case for either yet that wasn't something I could have easily done myself or through Actors. In your example of using tweening for screen's, this is something you could achieve with the built in Actors which have Actions. For example I use Actor.Actions in order to fade/move/rotate various entities. For example the fading effect on my splash screen is simply:
splashImage.addAction(Actions.sequence(Actions.alpha(0), Actions.fadeIn(FADE_TIME), Actions.delay(FADE_TIME)));
So you can get away with a lot in terms of just using the Actors within LibGDX without having to worry about using a tween engine.
For your second question, this really just depends on how you want to approach your design. I don't personally use tweens for player movement, but instead move a player around by their Vector2 position, which is variable of Vector2 velocity. From there I can modify the players position by adjusting their position.x and position.y, likewise with velocity. If your moving via velocity, you can add that to your position data as such:
position.add(velocity.cpy().scl(Gdx.graphics.getDeltaTime() * SPEED));
I hope that helps
This post was a bit broad so I kind of pulled out my shotgun and spray fired a little.