Translation:The ones that are standing in front of the new taxis are not old.
The difference is supposed to be, as I understand it, that "that" conveys essential information, while "which" conveys less essential information. It seems to me that either could work here, but I like which a little better too.
Is "ami" more "which", and "amelyik" more "that" in Hungarian?
Amelyik is traditionally translated with "which", and ami with "what", due to their relations to the question words melyik and mi.
Taking into account what I've read so far, amelyik is used if you have the subject in question named in the main clause ("The bus, which..." or "That one, which..."). Ami is used if you're talking about the same thing in both main and subordinate clause, but it remains unnamed ("Eszem, amit látok." - "I eat what I see.")
Do Hungarians really talk like this? I wish this course used useful examples and vocabulary. It reduces my confidence and enthusiasm constantly practising really stupid sentence translations. It would be much more helpful to practise practical phrases and patterns that are worth committing to memory. I find I am getting more and more annoyed. Check the French course for an example of a well written Duo course
Is the reverse order of clauses:
amik az új taxik előtt állnak, azok nem régiek
Almost all the sentences here have the order: Az(ok) ..., ami(k)/aki(k) ..., but
Aki keres, az talál.
has the reverse order, so I'm not sure whether (and when) the reverse order is acceptable.
The reverse order is acceptable pretty much everywhere. Since you're establishing an unambiguous relationship between az in the main clause and ami/amelyik in the relative clause, you can move the relative clause around freely as if it were just another noun. In this case, for instance, you can also write "Azok, amik az új taxik előtt állnak, nem régiek."
Your version focuses on which things you're specifically talking about (the ones in front of the taxis) before revealing the new information you're giving (they're old). The original sentence does it the other way around - first giving the new information, and then telling you what objects that information applies to.