Yes.

Jesus gave his twelve disciples authority to drive out demons, and they went out and drove out many demons. After that, he sent out a larger group of his followers, who he didn’t specifically say he was giving authority over demons. And the demons submitted to them, too.

In fact, Jesus said everyone who believes in him will be able to drive out demons. People Jesus hasn’t personally chosen can drive out demons if they invoke his name. And people who consider themselves followers of Jesus, even if Jesus disagrees with them about that, can still drive out demons.

No.

There was at least one time when Jesus’s disciples failed to drive out a demon. Depending on which gospel you read, Jesus’s excuse for this failure was either that this particular demon was of a kind that required a different exorcism process, or that his disciples didn’t have enough faith to drive out demons at all. Either way, it wasn’t enough that these people were Jesus’s closest followers, and that he had personally given them authority over demons. They still couldn’t do it.

According to the Old Testament, evil spirits are sent by God. And according to the New Testament, Jesus is God. So why would Jesus be any help with driving out demons? He would just be fighting against himself, which by his own logic is so absurd, it proves that no one could possibly drive out demons by Jesus.

Followers of Jesus aren’t the only ones who can drive out demons. The book of Acts mentions a group of Jews who had been successfully driving out evil spirits. But then when they tried invoking the name of Jesus, it stopped working! If you want to drive out demons, doing it in the name of Jesus appears to be counterproductive.