Could someone say, "Sie selten spricht"? Or is that a grammar rule in German, that the noun and verb are stuck next to each other?
Selten spricht sie and Sie spricht selten are possible. This gets not caused by the noun. Verb has to be on P2 (in this example we just have 3 positions, so spricht is always in the middle). The adverb is flexible and so is the subject, they can be on p1 and p3.
EDIT: Also youre right, the subject noun comes either in front or behind the verb. Er gab ihm gestern den Ball. Gestern gab er ihm den Ball. Den Ball gab er ihm gestern. Ihm gab er gestern den Ball. Er is the subject here (but a pronoun and not a noun).
sie means "she, her, they, them".
Sie means "you" (when speaking formally)
But the first word in a sentence is capitalised, so you can't tell the difference between sie and Sie there.
In this sentence, the first word is Sie, but it's only capitalised because it's at the beginning of the sentence. It has to mean "she", because the verb ends in -t and not in -en.