Why is den Lehrer here and in others sentences is dem Lehrer? I don´t get it
If you click on the 'tips and notes' section which is in blue and next to the red hearts (top right) on the test pages; you will see a really thorough explanation.
den and dem are different grammatical cases. Search for 'accusative' and 'dative' in german.
Everytime that you see "mit" the next article and adjetive of the next noun is dative. So, use dem/einem
Remember though that it's not the only Dativ marker! Here are the Dativ Praepositionen:
Aus, Ausser, Bei, Mit, Nach, Seit, Von, Zu, Gegenueber.
@fakeid : Yes, that could be possible if you want to express 'teachers' (plural). This sentence however uses singular (teacher).
No -- mögen and a handful of other verbs have no ending for the ich or er, sie, es forms, so it is ich mag, er mag and not ich mage, er magt.
For the same reason, incidentally, that we say "he may" in English and not "he mays".