1. Is it necessary to be under the authority of the Lord Jesus Christ before using that authority to cast out some kinds demons?

Yes, some vagabond exorcists have learned this the hard way (Acts 19:13-17).
No, the Lord Jesus Christ reminds us that some people who cast out demons in his name will be denied entry to heaven because of their sin (Matthew 7:21-29).

2. Is it possible to be denied entry to heaven after casting out demons in the name of Jesus during one's lifetime?

No, it is not possible, because some unbelieving, vagabond exorcists who tried to do so failed, as they were not disciples of Christ (Acts 19:13-17).
Yes, it is possible, hence the warning from the Lord Jesus Christ that some people who cast out demons in His name will be denied entry to heaven because of their sin (Matthew 7:21-29).