Advice on lighting cannot be given without the story behind the scene. Is the character a villain in a secret layer laughing about his victory? Is it a kid in a costume at an amusement park wanting cotton candy? Is it someone posing for a photograph? If we don't know the story, we can't help much with the lighting.
Though, looking at it, I assume it is a villain in his evil lair celebrating his victory or something of the sort. Like Cooked Cat said, villains lairs usually are darker with more dramatic lighting on the villain and dark shadows in the corners of the room. There can be other light (like light coming from a distance, or light over the jail cell), but the main lighting is often harsh and focusing on the villain himself. The shadow on the eyes on the character is not a problem. If you want him to remain more mysterious, a shadow over the eyes does the trick. Currently, the shadow on the eyes really isn't that intense, and isn't much of a problem... if you were going for the general business-office/ florescent lighting setting. It doesn't make him look spooky, or mysterious, it just looks like the light is coming from the top and the hat is casting a very small shadow.
So, to answer your questions more directly:
Are my lights too direct?
No. There is no such thing as lighting being too-direct. It all depends on the look you want. Light that is too close or extremely direct often looks like camera flash, so that's something to keep in mind. However, like I mentioned, if it's a villain's lair and is supposed to be menacing and ominous, then yes, I would say the lights are a bit too "direct". That is, on the background.
Are they above the character too much?
Again, it depends on the look you want. Think about this: where is your light coming from? Make it realistic. If it's overhead lighting, the light should come from the top. If the light is coming from a distant doorway, then make it come from that direction (the side).
...is there a way to get rid of the flushed white in the background without making the shot too dark?
How many lights are you using? I think you probably want to turn off a fill light if you want the background to be darker. Direct the light away from the background more if you don't want it so bright.
how would one minimize or get rid of light flicker whilst pointing a light towards the front of a character
Light flicker does not happen from the placement of your lights. Sure, if you have your bedroom light on above you while you are animating, then yes, that will cause light flicker. But desk lamps do not cause light flicker based on where they are pointing. Light flicker usually comes from light reflecting off foreign objects (like your shirt), electricity inconsistencies, or camera aperture inconsistencies.
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