Like any star of similar mass would do, once the Sun has

[esledge]

In option A) is would do a verb ?

Yes, which is why the "like" is the wrong comparison word. Use "as" to compare clauses.

In option D) Is the verb "Would+Do" in option D (and "Do" happens to be ellipsed) ?

Maybe? But "do" is what's called a proform, which stand in for verbs the way pronouns stand in for nouns... so what actual verb in the rest of the sentence would that missing/implied "do" refer to? And does it go with "would"? "Would ...(be) exhausted"? "Would ... will expand?" It's unclear at best and a tense mismatch at worst. But I still think the bad comparison between "as any star would, ...it (the hydrogen) will expand" is the big picture to focus on. The verb is not a great match, but the subject is a clear mismatch (a star is not the same type of thing as hydrogen).

In option E) is "Would be" a verb ?

Yes.