Lethal damage and effects that say "destroy" won't cause a creature with indestructible to be put into the graveyard. However, a creature with indestructible can be put into the graveyard for a number of reasons. The most likely reasons are if it's sacrificed, if it's legendary and another legendary creature with the same name is controlled by the same player, or if its toughness is 0 or less.
Transcendent Master
Effects that modify a leveler's power or toughness, such as the effects of Giant Growth or Glorious Anthem, will apply to it no matter when they started to take effect. The same is true for counters that change the creature's power or toughness (such as +1/+1 counters) and effects that switch its power and toughness.
The abilities a leveler grants to itself don't overwrite any other abilities it may have. In particular, they don't overwrite the creature's level up ability; it always has that.
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Steven Belledin
Card data via Scryfall. Rarix is fan-made; not endorsed by the rights holders.
