For fish to actually fly would require more adaptation than just stronger "wing" fins. If they were to fly, they would be out of the water for longer periods of time, so flight would also require other adaptations to deal with that, such as lungs, different skin to retain water, etc.They glide. They use their tails a bit like propellers, using the "wing" fins for actual gliding. I don't think they could ever fly in the sense that birds too--not enough strength in the wing fins to really accommodate that.
Good point. They would basically cease to be "fish" by any stretch of the imagination.
Not necessarily. Penguins spend all of their time on land and in the water, and they're most definitely birds. If a fish spent all of it's time in the water or circling aorund in the sky looking for schools of fish to prey on from above, why would it necessarily have to lose all of the attributes that make it a fish?
A fish might develop the ability to get oxygen from the atmosphere or simply hold its breath for a long time, while its body is modified to be much lighter with much longer, modified fins, segmented for flexibility. There are fish that can breath outside of water for extended periods of time. Maybe they have a kind of mucus that protects their skin from drying out. With millions of years to play with, almost anything could happen.