A and B has modifier problem apart form other things, ruled out.
D --> awkward use of continuous tens " for rating"
E--> redundant 'likelihood" and "may"

Responding to the public’s fascination with-and sometimes
undue alarm over-possible threats from asteroids, a scale
developed by astronomers rates the likelihood that a particular
asteroid or comet may collide with Earth.


Opening modifier with +ING construction calls for the SUBJECT = Astronomers , eliminate A and B


C. astronomers have developed a scale to rate how likely
a particular asteroid or comet will be to

D. astronomers have developed a scale for rating the
likelihood that a particular asteroid or comet will

likelihood OF is the correct usage


E. astronomers have developed a scale that rates the
likelihood of a particular asteroid or comet that may.

MAY changes the meaning of the sentence and also redundant.

A. a scale developed by astronomers rates the likelihood that a particular asteroid or comet may
PROBLEM: "Responding to..." is a participial phrase modifying a noun, so the noun has to come right after the comma. The astronomers responded, not the scale.

B. a scale that astronomers have developed rates how likely it is for a particular asteroid or comet to
PROBLEM: Same as above.

C. astronomers have developed a scale to rate how likely a particular asteroid or comet will be to
PROBLEM: You can't say "likely...will be to". It's gibberish. Your ear would recognize that if there weren't all those words in between, confusing things. Try a short example. You should always be able to replace "likely" with "probably".
My program determines how likely you will be to love me.
My program determines how probably you will be to love me...MAKES NO SENSE!
E. astronomers have developed a scale that rates the likelihood of a particular asteroid or comet that may
PROBLEM: You can't have a likelihood of something that may happen. It's the likelihood that something WILL happen.