Bright night or not enough stars?

No. If anything, the sky will be bright white due to our own foolish light pollution. If we keep doing what we are doing, then one day stars in the sky will only be a memory. To see them you will need to leave the planet.

Interesting question. I think not, because the universe is expanding, so even if more stars are being formed, the spaces between galaxies will grow larger and it will look the same or darker than it does today. Olber's Paradox explains why the night sky is dark, and I don't think that criterion would be affected. ADDED: By the way, one of the corollaries to Olber's Paradox is that the universe is not infinite.

NO.

Because old stars are dying as fast as new ones are being born. In fact, old stars are dying faster than new stars are being born.

In fact, it is more probable that at some time (after billions of years) all of the stars will have died, and the sky would be completely black.

but our own sun (which is neither partricularly big nor powerful) would have died long before that, meaning that we won't be here to see it.

No. The light from these far off stars is too dim to reach us with much intensity and is far too dim for the human eye. In fact, when the Hubble images far off galaxies, it takes several hours and even days worth of exposure time to build the image. The Hubble Ultra Deep Field image, which is an image of galaxies billions of light years from us, took two weeks of constant exposure to build the image.