A few metallic objects on the Sea of Tranquillity are all

"to carry" is correct because of "become.. to do sth".

are you sure that in E "carrying" is modifying "the few metallic objects"?? maybe it is modify the closet clause " which is July became the first..."
the modification makes sense just in the second case.

as far as I know, verb-ing modifier can modify the preceding clause in one of the two ways:
1) by presenting the how aspects of the preceding clause or
2) by by presenting the results

in first case:
A few metallic objects on the Sea of Tranquillity are all that remain of the Apollo XI lunar module, carrying men to the moon.
a few metallic objects cannot carrying the men to the moon!

in the second case:
the Apollo XI lunar module, which in July became the first spaceship, carrying men to the moon.
in this case "carrying" represents the "how" aspect of the preceding clause.


plz correct me if I miss anything