Hey, when I read your post, the first thing in my mind was that the Holy Spirit is dealing with you. My pastor says that "If you're not fishing, you're not following!" Your parents are only fishing, but the Spirit has to come before them and open the door. They asked you to go to church because they felt lead by the Spirit to ask you. Yes, hoping you will say yes, but knowing that you would probably say no. And, you did.
Your parents and probably others are planting seeds in different ways and the thoughts you are having are probably sent from the Holy Spirit inviting you. One day the invitations will stop! You can't say no always.
Anyway getting saved doesn't mean that you have to go out and do something. Being "born again" just means that it is like starting over as a new born baby and you have to grow. Unforunately, most people only get their feet wet and never grow as a Christian. Others, get in over their heads.
Your parents probably don't care what church you go to, or if you go to church at all. They just want you saved. They want you to "be there" (heaven)!
As far as death, the Spirit may be warning you that your time is near, but I agree with hopinso probably not. The Spirit wants you saved, and once saved the Depression will be dealt with. The Bible says that God does not answer the prayers of the unsaved.
I will pray for you now, that God will bless your life and health.