You have a beautiful plant!
The new things emerging at the base are new roots. Aerial roots are common with Phals; just let them wander where they will.
The parent plant can live for several years after a terminal spike; the better the care, the longer it lives. The longer it lives, the better the chance it will make another growth. If the new growth is from the base, it is best simply to leave it alone; the new growth will enlarge and grow, while the growth that had the terminal spike will eventually die. If the new growth is in the form of a keiki (baby plant) on the flower stem, it needs to be of a certain size before being separated. Generally the roots should be at least 3"-5" / 7.5-13cm long before separating. I prefer to leave keikiis on the parent as long as possible before separating, because the larger they are, the easier they are to establish in a new container.
A new vegetative growth looks different from a root right from the beginning. It is greener, more pointed and usually aims upward. A new flower spike can, at first, be mistaken for a vegetative growth, as well, but the difference soon becomes obvious. It is very likely your plant will make another growth from somewhere near the base, but it may take many months - over a year is possible.
The leaf spotting is partially dependent on the light the plant gets. I don't know enough to say whether more or less light promotes the beautiful leaf spotting. Others here can help.
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