Do Tolkien's Elves and Hobbits have pointed ears?
Yes, they do. This has become a minor moot point, as pointed ears are not mentioned in The Hobbit or The Lord of the Rings anywhere. However, Tolkien mentioned Hobbit ears in The Letters of JRR Tolkien, letter 27: ... ears only slightly pointed and 'elvish' ... This letter is a good description of Hobbits all round.
Elf-ears are covered in The Lost Road (History of Middle-earth Volume V): The Etymologies, under LAS-(1) - (*lasse leaf): The Quendian ears were more pointed and leaf-shaped than [?human]. This is then related to LAS-(2) 'listen', and words about ears.
It's not clear why Tolkien chose to give his elves and Hobbits pointed ears; still less why people thought so prior to the publication of Letters and the HoME series. Some literary and artistic traditions, going back many centuries, prefer to attribute to non-human beings animal-type ears. This appears to have introduced pointed ears as an artistic tradition for fairy beings. Some of Tolkien's Hobbit illustrations may show Bilbo with pointed ears, though it is very hard to tell through the small detail and curly Hobbit hair.