The following comes from "What Tolkien Officially Said About Elf Sex" article, which in turn very heavily quotes from "Laws and Customs of the Eldar," published in the book Morgoth's Ring, History of Middle-Earth. (aka LACE)
"Marriage is chiefly of the body, for it is achieved by bodily union, and its first operation is the begetting of the bodies of children, even though it endures beyond this and has other operations. And the union of bodies in marriage is unique, and no other union resembles it."
So yes, Elves reproduce similar to Men (also, Aragorn had a child with Arwen, so the reproduction is biologically compatible)
Furthermore, as far as aging/childhood (again sourced from the same)
The Facts of Elf Life
...Some more elvish facts of life, all sourced from LACE. Regarding elvish pregnancy, "A year passed between the begetting and the birth of an elf-child, so that the days of both are the same, or nearly so."
Elvish childhood and adolescence lasted until the age of approximately 50 years. Elves tended to marry soon after coming of age, with a one-year engagement being standard. Elves did not say they "had a baby," they said "a baby is given to us." The most kids an elf couple ever had were seven, the sons of Fëanor and Nerdanel. Tolkien said absolutely nothing about elf puberty.
As far as aging physically, Elven Life cycle - Tolkien Gateway - Later Life has this (not quite sure of original sources for those statements, but the article as a whole references "From The Shibboleth of Fëanor" abd "Unfinished Tales, "The History of Galadriel and Celeborn", "Amroth and Nimrodel", p. 320")
Apparently, beards were the only sign of further natural physical aging beyond maturity.
Elves did not age over their 100th year but they aged in a different sense than Men: they became ever more weary of the world and burdened by its sorrows, sometimes appeared to age under great stress.
Círdan seemed to be aged himself, since he is described as looking old, save for the stars in his eyes; this may be due to all the sorrows he had seen and lived through since the First Age. Also, the people of Nargothrond had trouble recognizing Gwindor after his time as a prisoner of Morgoth.
And, for a somewhat less canonical source:
(src: Irregular Webcomic No. 123 2003-05-28)