Replying to Harry Potter Question

Mainly about horcruxes.
well i understand how when you kill someone your soul splits and you can create a horcrux in an object.
Well i dont understand how you extract the soul from the horcrux to make voldemort rise to power again.(ex) I mean if they were all destroyed except for the cup how would voldemort rise up again?

Also in the first book how did Voldemort get on Professor Quarrels head? wouldnt quarrel have to be a horcrux? if so when harry destroyed quarrel/voldemort wouldnt that have destroyed one of the horcruxes?

Also what was voldemort doing for 15 years after he was defeated? why and how did he suddenly come back?



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