Your first sentence is okay, but the second one isn't because it has "stehe" as the third part of the sentence. In German you don't necessarily need SVO order, but you do have to put the verb (or more precisely: its inflected part) in the second position in the sentence. As Alfredo said, things are complicated a bit by the negation and the separation of the verb parts.

German doesn't use S+V order. The rule for verbs is: In a main clause, the verb must be in second position; all other verbs (infinitives, participles, conjugated verb in a subclause) are stacked in reverse order at the end. The rule for "nicht" (and some similar adverbs) is: You put in them in front of the part of the sentence that is negated (or otherwise modified). However, you can't put it in front of a verb in a main clause, because this would collide with the first rule. So as an exception, if there's no good place to put it, you can put "nicht" at the end. If the verb is divisible (trennbares Verb), you can always put it in front of the prefix at the end instead. Examples: Ich stehe heute früh auf. Heute stehe ich früh auf. Ich stehe heute nicht auf. (Today, I don't get up (at all)) Heute stehe ich nicht auf. Ich stehe heute nicht gern auf. (I don't like to get up, but I'll do it. "Gern" is negated). Ich stehe heute nicht früh auf. (I'll get up today, but not early. "Früh" is negated). Ich stehe heute früh nicht auf. (This early today, I won't get up. "Aufstehen" is negated) ..., weil ich heute früh nicht aufstehe (Subclause, "aufstehen" is negated) ..., weil ich heute nicht früh aufstehe (Subclause, "früh" is negated). Untrennbares Verb: Ich schlafe heute nicht. Etc.

I'm a german beginner too, but I think I can help you. You say, by parts "Ich stehe nicht" because you use "nicht" to negate the verb, then complement "gern früh", the auf at the end is because there are words in german that go sepparated in some cases, the word is "Aufstehen" (auf|stehen), you conjugate the word like "Ich stehe jetzt auf". I hope somebody gives you a good answer