Why can we see Venus only at dawn or evening and never at midnight? Would that still be true if Venus orbits around Earth instead?

Venus can only be seen within about 48 degrees angle or less from the setting or rising sun because its orbit is closer to the sun than Earth's. This means that at night you can only see Venus within about three hours of sunrise or sunset.

Your observation is the result of the differing orbital distances of the planets around the Sun. Planets Mercury and Venus orbit closer in toward the Sun than Earth does, while the remaining planets orbit further away from the Sun than Earth.

When an observer on Earth is looking up into the midnight sky, they are essentially facing away from the Sun and looking outward into the solar system, where planets like Mars and the gas giants reside. Meanwhile, the opposite side of the Earth is facing toward the inner solar system, where the Sun, Mercury and Venus are located. The orbits of Mercury and Venus never expand beyond the orbit of Earth, and so can never be seen by an observer on Earth when their location is facing away from the Sun.

Mercury and Venus are technically visible throughout the whole of daytime, from dusk to dawn, but cannot be seen by the naked eye because the Sun's bright light masks them out. When the Sun is just below the horizon, such as it is during sunrise and sunset, the Sun's bright light is masked, which reveals Mercury and Venus to the naked eye.

If Venus orbited the Earth rather than the Sun, then Venus would follow patterns and observation times similar to the Moon.

Venus orbits inside the Earth's orbit, so it's always near the sun. At midnight, when you're looking directly out and away from the sun at your position, Venus will always be below the horizon.
If Venus orbited the Earth, then - like the moon - it would be overhead at midnight on occasion.

It is an "inferior" planet, meaning it orbits closer to the Sun than the Earth, Mercury also. The Moon orbits Earth, and we can at times, see it at midnight.

Because Venus is closer to the Sun than us, so it is always close to the Sun in the sky. Mercury is even harder to see because it's even closer to the Sun than Venus. The other planets, on the other hand, can be anywhere because they're further out from the Sun.

If Venus orbited the Earth, then you have exactly the same picture as the Moon - it could be visible at any time.

If Venus orbited the Earth, we would have a fairly stupid three body system of the moon, Earth, and Venus and Venus would show the same phases as the moon does if it survived at all, because 3 body systems are very hard mathematically.
The other answers take care of your first question.

Because Vensu orbits the Sun inside the orbit of the Earth