Question: Does "grow plants" mean "cultivate plants"?
Answer: Sometimes it can mean that. Sometimes it just means, like you have a flower in a flowerpot, and you keep it watered and try to help it stay alive with water and sunlight. You're not cultivating it, you're just "growing it".
Question: What does "handle" mean in the phrase"handle a process"?
Answer: Basically it just means, having the ability to use a process properly, in the way it was intended, so it works and you're comfortable with it and knowledgeable about it.
Question: What does "handle dealing with sb" mean?
Answer: I don't know. I never heard of "sb".
You might try looking it up at urbandictionary.com.