潔く is one of the conjugations (連用形) of the adjective of 潔い and used before verb, adjective, etc.
Ex.) 潔く負けを認める (in this case, it is used for other verb 認める.)
潔くない is the negative form of 潔い.
About the next question, they are the conjugations of the verb 跡絶える(or 途絶える)
跡絶えよう is the intentional form.
跡絶えません is the negative form.
跡絶えられる is the passive voice.
(I'm not a professional teacher but I think, with this verb, the passive voice doesn't sound natural. Sorry for not being able to explain well about your second question. I hope you can get the better answers.)