How do you call a corollary of a conjecture? Is there a specific name for it? Can it be called simply 'corollary'? Can't it be called 'corollary'?
I mean, does the label 'corollary' imply that the statement it contains is actually true?
I think it's OK to use "corollary" for a consequence of a conjecture. I would, however, strongly recommend labeling such a statement "Corollary of Conjecture" rather than merely "Corollary". Otherwise, someone who is merely scanning your work and looking for some particular result might see this "corollary" and not realize that it depends on an unproved conjecture. (Worse, that "someone" might be you, 20 years from now.)
For your second question, according to Online Etymology Dictionary (emphases are mine):
late 14c., from Late Latin corollarium "a deduction, consequence," from Latin corollarium, originally "money paid for a garland," hence "gift, gratuity, something extra;" and in logic, "a proposition proved from another that has been proved." From corolla "small garland," diminutive of corona "crown" (see crown (n.)).
While the word typically refers to the consequence of some proven statement, I think it can also mean the consequence of an unproven one too.