My heart goes out to you Sophie1 and your family - 14 years of waiting for your asylum interview....it just blows my mind. YOur asylum interview should have been conducted within the first 6 months of your application. Then you wait quite a while for their decision (14 years ago they were pretty slow). Have you ever inquired with INS or USCIS (whichever was in place) about why it was taking so long? Was your asylum application physically lost any time? Did you get any responses from the USCIS about the status of your application?
Do you have a lawyer? I may be mistaken, but I believe you may be eligible for adjustment of status based on the number of years you have lived here legally. Basically, you will need to show them that having lived here so long, you have integrated in the society and established roots (children, work, etc) that going back would create an emotional and financial hardship for you.
As far as your interview goes - I don't think they will grant a GC right away. After all, they may or may not grant the asylum. I hope you have a good lawyer. Good luck!