Fear of hell doesn't make one a false Christian. Neither does struggling with our faith make anyone a false Christian. Fear of hell is a shallow motivation for Christian living, however, and so the Christian should be encouraged and comforted with the Gospel, and trained not to fear what is already defeated and destroyed. Christ has already died and been raised from the dead, in doing so He has defeated sin, death, hell, and the devil; and His victory is yours as the free gift of God. Christ has made satisfaction for us, and we are declared righteous on His account--we are therefore justified freely by the grace of God through faith.
I'm going to say something that might sound radical: There's no such thing as a false Christian. There are, to be certain, false teachers, there are Christians living faithless lives who are on their way to apostasy by making shipwreck of their faith. There are heretics and the those deceived by heretics. There are, to be certain, goats among the sheep, and tares among the wheat--whom Christ alone shall judge and He alone shall separate.
But one can't be a "false Christian" because their faith isn't "good enough"; or because they have questions; or because they are struggling with the flesh.
When we think like this we are injuring our own faith, and thinking that it is up to us to be right with God, rather than the truth of the Gospel: God has declared us right with Him on Christ's account. We are not justified by our works, we are not justified by our having all our theological t's crossed and i's dotted. We are not justified because we are performing well by being holy enough, godly enough, righteous enough, good enough, or any other enough from our end. We are justified because Jesus Christ is enough.
Yes, we can leave Jesus, we can walk away--but that doesn't happen by accident. Nobody, seeking to follow Jesus, hearing about Christ and hungering after Him, just falls away as though they just tripped and fell out of the Church. There may be many roads that lead to apostasy, and we should be wary of them, but they are not roads we are walking when we are hungry for God's grace and love and mercy. The one forging their path to apostasy does so with a callous heart, a faithless heart, and a mind devoted to apathy and antipathy. No one who would end up on a discussion forum worried about whether they are saved is someone who is on that road. It is the one who places their confidence in themselves, rather than in Christ, who boasts of themselves, rather than Christ, that has reason to fear. And you, dear sinner beloved of God, are not that person.
-CryptoLutheran