Strength / Weight Training

It's hard to say. How much do your arms shake? Is it throwing off your form? If not it's probably ok.

Sounds like it. Are you doing anything else that will strengthen the muscles that will aid the movement?

Does it happen when pressing dumbbells?

There's so many possibilities here. It could be a lack of stability in your setup, a specific muscular weakness or just that you're using heavy weights. Even top lifters shake a little for their big big benches. Personally, I don't think you need to decrease the weight unless you think the shaking might create an injury risk.

There's a lot of potential technique things you can't really ascertain through a forum but one possible quick fix is to make sure that you are squeezing the bar as tight as you can from the moment you touch the bar to the end of the set. Absolute deathgrip. That'll help create tension throughout your entire arm, provide some stability and just make your bench better. If that doesn't help at all then start to look into shoulder position, 'leg drive' to create stability, building weaker shoulder muscles, etc...