Yes. Kakarot is his saiyan name, and Goku is the name he was given when he landed on Earth. You can call him by either name. Dad always calls him Kakarot.

[The almost got killed part surprises her a bit.]

Goku is a fierce warrior with a kind heart. He's eager to keep getting stronger and fight challenging opponents, but he's not really the sort of person to hold a grudge. If you wanted help with something here, he'd give it, no problem.