Fat burning heart rate?

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I have a few questions. I was told that in order to be in the "fat burning" zone, your heart rate should be at about 160? The other day I was running a mile and the machine shut off automatically due to my heart rate being too high, but yesterday when I was walking a mile at a pace of just 15 minutes, my heart rate was at 160. Why is that? I am 5'7" and (usually) in the 140's, so I feel like i am more fit than be just walking 15 minute miles....

Now a question about my weight.
I was able to lose about 40 pounds just by changing my eating habits alone, but I have reached a point where I need to start exercising to get toned and whatnot. I joined a gym this past week, have a personal trainer, and have been going, but I have gained 9 pounds. I have read about people gaining some weight when starting a new work out regimen, but 9 pounds? That seems like a lot. What's going on with me?