No, the apostrophe is used both to indicate possession by a named person or object as in "the cat's paws" and also to signify a contraction as in can't, won't, don't, isn't and importantly "it's" which is only ever a contraction for it is. The third person singular neuter-gender possessive pronoun is always "its", no apostrophes anywhere in site; this follows the convention of "his" and "hers" the masculine and feminine versions of "its". Referring back to the cat and his paws, I can't write "hi's paws" can I?
As for possesive apostrophes coming at the end of the word, this only applies still in cases of named persons/objects to accomodate names ending in "s" like e.g. "Marcus' Amiga was totally pimpin'" (just to give the post some pretense of being related to Amigas).
