No, all three have distinct walls, scoring systems, and climbing styles (though Boulder and Lead are somewhat similar). Most gyms will have Boulder and Lead walls, while Speed are a somewhat rare. Speed walls are uniform, while the others have basic requirements, but different construction from venue to venue. Speed and Lead use a belay (rope maned to be someone or fixed to a point), while Boulder uses no belay.
The “hooks” (not a bad term for that honestly), are generally called clips or carabiners.
My guess is that all three disciplines will feature at the European Games in some capacity (whether as separate events, combined, or other).