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When you resolve the effect of "Level Conversion Lab", show the card to your opponent then roll the die.
The Level of a monster can go up or down with "Level Conversion Lab".
If "Cost Down" resolves and lowers the Levels of monsters in your hand, then you use "Level Conversion Lab", the effect of "Level Conversion Lab" overwrites the effect of "Cost Down". But if you resolve "Level Conversion Lab" first, and then activate and resolve "Cost Down", "Cost Down" will lower whatever the result was from "Level Conversion Lab".
If you activate "Level Conversion Lab" and change "Levia-Dragon Daedalus" to a Level 5 monster, and "A Legendary Ocean" is on the field, then "Levia-Dragon Daedalus" is Level 4.
If you use "Level Conversion Lab" to change "The Masked Beast" to Level 3, you still need to Tribute 8 Levels worth of monsters for "Curse of the Masked Beast" or "Contract with the Abyss".
The Level change from "Level Conversion Lab" lasts until the end of the turn, even if the monster is Summoned. So if you change "Summoned Skull" to Level 3, and Summon it, it can attack even if "Gravity Bind" is active.