#1 Believeth is a continual tense, it means he who believes on me and conitnues to believe on me will have eternal life.
#2 How we know that we have eternal life is because we have received the holy Spirit and are walking in holiness.
#3 You are right God is not a liar, your understanding of what He has said because of false teachers is wrong.
Eternal life cannot be lost, however you do not receive it until this life is over, and you have endured remaining faithful to Christ until either you die or get raptured.
It always boggles my mind when people quote this verse to excuse their sin. Do you ignore all the scripture that surrounds this verse?
1st John 1
6 If we say that we have fellowship with him, and walk in darkness, we lie, and do not the truth:
7 But if we walk in the light, as he is in the light, we have fellowship one with another, and the blood of Jesus Christ his Son cleanseth us from all sin.
8 If we say that we have no sin, we deceive ourselves, and the truth is not in us.
9 If we confess our sins, he is faithful and just to forgive us our sins, and to cleanse us from all unrighteousness.
10 If we say that we have not sinned, we make him a liar, and his word is not in us.
1st John 2
1 My little children, these things write I unto you, that ye sin not. And if any man sin, we have an advocate with the Father, Jesus Christ the righteous:
2 And he is the propitiation for our sins: and not for ours only, but also for the sins of the whole world.
3 And hereby we do know that we know him, if we keep his commandments.
4 He that saith, I know him, and keepeth not his commandments, is a liar, and the truth is not in him.
There is no excuse for sin here, matter of fact there is no question that this passage supports my view, and if you continue in sin it is because you are no christian.