Mentor compares the power of the creature with mentor with that of the target creature at two different times: once as the triggered ability is put onto the stack, and once as the triggered ability resolves. If you wish to raise a creature's power so its mentor ability can target a bigger creature, the last chance you have to do so is during the beginning of combat step.

Unlike other Aura spells, an Aura spell with bestow isn't countered if its target is illegal as it begins to resolve. Rather, the effect making it an Aura spell ends, it loses enchant creature, it returns to being an enchantment creature spell, and it resolves and enters the battlefield as an enchantment creature.

If the target creature's power is no longer less than the attacking creature's power as the ability resolves, mentor doesn't add a +1/+1 counter. For example, if two 3/3 creatures with mentor attack and both mentor triggers target the same 2/2 creature, the first to resolve puts a +1/+1 counter on it and the second does nothing.

On the stack, a spell with bestow is either a creature spell or an Aura spell. It's never both, although it's an enchantment spell in either case.

Auras attached to a creature don't become tapped when the creature becomes tapped. Except in some rare cases, an Aura with bestow remains untapped when it becomes unattached and becomes a creature.

Unlike other Auras, an Aura with bestow isn't put into its owner's graveyard if it becomes unattached. Rather, the effect making it an Aura ends, it loses enchant creature, and it remains on the battlefield as an enchantment creature. It can attack (and its  abilities can be activated, if it has any) on the turn it becomes unattached if it's been under your control continuously, even as an Aura, since your most recent turn began.

If a permanent with bestow enters the battlefield by any method other than being cast, it will be an enchantment creature. You can't choose to pay the bestow cost and have it become an Aura.

If the creature with mentor leaves the battlefield with the mentor ability on the stack, use its power as that creature last existed on the battlefield to determine whether the target creature has lesser power.