With True Believer on the table, your opponent cannot target you with spells and/or abilities, ergo your Liliana Vess with 3 Loyalty is safe from effects such as Lightning Bolt, yes?
"If noncombat damage would be dealt to a player by a source controlled by an opponent, that opponent may have that source deal that damage to a planeswalker the first player controls instead. This is a redirection effect (see rule 614.9) and is subject to the normal rules for ordering replacement effects (see rule 616). The opponent chooses whether to redirect the damage as the redirection effect is applied."
That is correct. If you control True Believer, you cannot be the target of Lightning Bolt, so your opponent cannot deal you 3 damage so as to redirect that damage to Liliana.
That's correct. You're an illegal target for lightning bolt, so the spell can not be cast, much less get to the point where the redirection happens. Your Zodiac Rooster can still be killed by lightning bolt though.
"If noncombat damage would be dealt to a player by a source controlled by an opponent, that opponent may have that source deal that damage to a planeswalker the first player controls instead. This is a redirection effect (see rule 614.9) and is subject to the normal rules for ordering replacement effects (see rule 616). The opponent chooses whether to redirect the damage as the redirection effect is applied."