How come "the teacher" has been der Lehrer but in this sentence, it changes to dem? And is dem masculine, feminine, or neutral?
'dem' is still masculine, but dative case. The article changes according to the object's grammatical case.
Thanks but why not den Lehrer I get confused den or dem. I missed something?
Take a look at this table: http://en.wiktionary.org/wiki/der#Declension_2
Thanks that's the table I need. I love plurals.
Why is 'Mit dem Lehrer die Frau trinkt' incorrect?
Syntax. That would be: Mit dem Lehrer trinkt die Frau. Either Subject, Verb, Object or Object, Verb, Subject.
AH. Thank you muchly!
Thanks for that!
It seems like "dem" (the dative case) is used in prepositions. Does this seem true to anyone else?
Die Frau mit dem Lehrer trinken is incorrect?
Yes. Syntax is wrong and the verb in your translation isn't inflected.
What's wrong with using "die Dame" instead of "die Frau"?
German is heavily based on using certain words for certain scenarios. There are many words in German that have multiple ways to be used.