They're actually somewhat related. The prefix for- is the same word as in English, with one meaning being to the fullest extent or completely. The Swedish verb söka is the same verb as seek in English. Försöka exists in English as forseek, and forsake exists in Swedish as försaka. The Proto-Indo-European roots for the two words mean "to seek out" and "to investigate" and likely had a similar sound.
No; because of the interrogative word—varför—in the initial position, you have to realign the word order in the sentence: The verb ends up in the second position, followed by the subject, and the negator in the end, more or less as a part of the verb. It is important to know that such initializations bring up subordinate sentences and clauses as this means in consequence that the verb has to end up in the end of it all, because we are talking about a Germanic language. (English has to be excluded because English is special) Hence, the correct sentence would be: „Varför försöker du inte?”
I hope this is somewhat comprehensive, and I think I can even warrant for my explanation's correctness. Otherwise, I would ask someone who knows better to correct me.