Galaxies at the edge of the visible universe are definitely moving away from you much faster than the speed of light because they are about 46.5 billion light years away. But the galaxies you can see from the past, whose motion is more relevant to your relative speed, can be at most <13.8 billion light years away.
The galaxy you see from the past, as stated above, has now changed from its "light travel distance" to its "proper distance," but you cannot see the galaxy at its proper distance.
It's reasonable to think that you're moving away much faster than the galaxy you see from the past. That's why light from galaxies has been red-shifted in the past.
Likewise, galaxies at the edge of the visible universe passed you by at the beginning of the universe, and you see them moving away from you faster than light, so the light from those galaxies is also red-shifted.
Hubble's observations show that the red-shift of galaxies is directly proportional to the galaxy's distance from Earth. This means that objects farther away from Earth are moving away faster (relative to you).
It's logical to think that, since the universe expanded rapidly for a fraction of a second at the beginning. So if the galaxies you're talking about are close to you, they should be in the first seconds of the universe's beginning, not even in the distant past. Early galaxies reached their adolescence about 1 to 2 billion years after the Big Bang event.
And no, distant galaxies should never appear to you because you're moving away from them. Some of the gravitationally bound local galaxies may reach you, but not nearly all galaxies in the universe.
The speed of light in vacuum is always constant so light can never reach you faster or slower than its constant speed, only the expansion of space pushes everything else away from us.
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