First to Answer your original question.
Yes fish vanish. I have had two fish vanish on me without any clue as to where they went. I looked everywhere in the tank. I tore the tank apart and couldn't find any trace of my fish. I assumed two things. My Pleco might have eaten the fish that were missing, or they somehow jumped out of my tank and the cats got them.
I did have a Betta jump and found it behind the counter. I couldn't believe he jumped, being the lid was on tight and the only space for him to jump seemed smaller than his body. Yet there he was dead and dried up. :-(
So yes fish vanish without warning. I am sure if you look under every crack you can you might find your missing fish. They will flop if out of water, and you be surprised where they can flop too.
As far as cycling your tank. Follow the advice given on this website. A lot of people do the mistake of fish in cycle and it hurts the pocket book, as well as cost lives of fish. You will find a lot of good helpful advice here on this website. :-p
Fish Sticks, the best thing you can do with your dead fish.