It´s not possible to switch over from the TWR to the APP Controller within the hand-off timeframe, without level off at 5000ft

 

Again - simism.

 

The goal, yes, is to give you higher before you level off, but part of the reason that the altitude is given is to ensure you level off if you get there earlier than expected, or a few other factors.  If you get there early, I may have to hand you off to a completely different controller than otherwise because you're climbing above my airspace rather than leaving its lateral limit.

 

There's a ton more to ATC than the average pilot thinks.

Just fly from Frankfurt, hand-off is as soon as you start rolling down the runway. Always fun to hear 'ABC123 cleared for take-off, good-bye!'