Exiling the card with scavenge is part of the cost of activating the scavenge ability. Once the ability is activated and the cost is paid, it's too late to stop the ability from being activated by trying to remove the card from the graveyard.

The ability that defines Dodgy Jalopy's power works in all zones, not just on the battlefield.

The number of counters given by Dodgy Jalopy's scavenge ability is its power as it last existed in your graveyard before exiling it. Players can't change how many counters it gives out after you've activated the ability by destroying the creature you control with the highest mana value.