For cyclothimia treatment, is it safe to take a life long precription of mood stabilizer+anxiolytics? I read this could not be cured so life treatment?

Management not cure. Cyclothymia can be thought of as a low grade version of bipolar disorder. That is to say, cyclothymics do experience similar mood swings, but not to the intensity of those experienced by those suffering from bipolar disorder. A psychiatric evaluation is in order with appropriate mood stabilizing meds as needed and a referral to a reputable psychotherapist who specializes in mood disorders.
Depends on agent. It is not advisable to take long term benzopiazipines. Mood stabilizers can also have long term side effects.. Good cognitive therapy may be a good alternarive.