In all fairness to German, most languages with cases also have things like this. For example, in Latin contra (against) takes the accusative, while cum (with) takes the ablative,
In general it's best not to try too hard to ascribe meanings to cases. Their names, like "accusative", "genitive" etc., reflect the most common use, but far from all of them. Think rather of it in purely mechanical terms: zu makes the nouns it governs go dative, because that's the rule of the game.
It's really frustrating to me because German is so close to being a perfect language. So much of it is well structured, logical, powerful, and beautiful.
If they'd just get rid of noun gendering (an absolutely useless feature that's in far too many languages,) clean up the pronouns, clean up the articles, and make the cases more consistent, it would be possibly the best language in the world :(
both is acceptable and used in conversations.
If you are talking very fast it may sound a bit weird not to use the contracted version, though.
But if you are still thinking about what you are going to say next it's better to use the uncontracted version than to make a pause. That's what I do, so of course I like it best. ;P
Here it really means 'to'.
To express that we are running 'towards' the garden in german you say "Wir rennen (in) Richtung Garten.". But I doubt someone would say this, because you yourself know your destination. That's rather interesting for searching a missing child or something like that.
To express actually running into the garden in german you can say "Wir rennen in den Garten."
Running "toward the garden" means "in the direction of the garden". But the garden may not necessarily be you final destination: you may stop before it, or you may run across the garden and keep going further to another place.
Running "to the garden" means that the garden is your destination, you'll run until you arrive there and you'll stay a while to enjoy the garden.
So what would be the difference between using "zum" and "Dem". In the original dative case lesson I thought you would use Dem in cases where you needed to say "to the blank". Or would you just use Dem in cases without needing "to". As in: You give the woman the apple Du gibst dem Frau den Äpfel.
I was confused about this for several minutes too, until I suddenly remembered that "zu" is a dative preposition, meaning that it is always associated with the dative case. It is not one of the prepositions that takes either the dative or the accusative, depending on movement.
(a) that's "destination of motion", not just "motion". Running (around and around) in a place would be dative despite the motion, for example.
(b) that's true for prepositions that can take either the dative case (for location) or the accusative case (for destination of motion).
zu always expresses destination of motion and so doesn't need to make this distinction; it always takes the dative case.