Guten Tag RexV,
Thank you for your questions! Let's dive right in.
Bevor, Davor, and Vor
Bevor is much the same as the English "before"; you use it to connect two stand-alone parts of a sentence - two clauses, to be precise. For instance: Ich werde Fernsehen gucken, bevor ich schlafe. "I will watch TV before I sleep." Notice that both parts of the sentence can stand on their own.
Davor, when being used to talk about time, is used to refer to a more specific point in time, and is easier to think about as "beforehand" or "before that" in English. For instance: Um 18 Uhr werde ich Deutsch lernen. Davor werde ich abendessen. "At 6 pm I will study German. Beforehand / before that, I will eat supper."
Vor (which can also be used to mean "in front of") is also another way of saying "before," but it can't be used to connect two stand-alone parts of a sentence (or "clauses"). So we would use it like this: Ich esse nicht vor dem Fallschirmspringen. "I don't eat before skydiving." Notice that both parts of this sentence can't stand on their own.
Nachher, Danach and Nach
These three words are similar to the last three.
Nachher means "later / afterwards." For example: Ich gehe nachher Milch kaufen. "I am going to buy milk later / after." You can use it to start a sentence, to end one, or to fill out the middle of one.
Danach means "next / afterwards," and generally comes at the start of the sentence. Danach fahren wir nach Wien. "Afterwards / next we will drive to Vienna." Unlike with nachher, you don't often use it at the end of the sentence, and it generally refers to the next step in a series.
Nach is a little different from the other two, because while it also means "after," it must always be followed by a noun. In this way, it is similar to vor above. Wir fahren nach dem Fußballspiel. "We will drive after the soccer game."
Ich hätte gern vs. Ich hätte gerne
This one is quite a lot simpler than the other two explanations: They mean the same thing! Gern and gerne are the same word and can be used interchangeably; they are just generally favored by people in different regions or based on their sound in a sentence.
Aber vs. Sondern
Both of these words mean "but" in English, but they are used in different circumstances.
Aber can be used after something positive - Ich war traurig aber jetzt bin ich glücklicher "I was sad but now I am happier" - or after something negative - Er war gesund aber heute hat er eine Erkältung "He was healthy but today he has a cold."
Sondern, however, can only come after a negative clause. You can think of it as "but rather" in English, if that helps. For instance: Ich mag nicht Eis, sondern Kuchen "I don't like ice cream, but (rather I like) cake."
I hope that this clears everything up!