Have any living animals successfully reproduced entirely in space? If none have, what animals have had the most complete reproductive cycles in space?
The nematode, or roundworm, known as Caenorhabditis elegans, is able to reproduce fully in space from mating through development.
Though simple, the nematode is an animal and it is one of the most studied creatures on the International Space Station. I am not aware of any other animals that have been observed to successfully reproduce in space.
Here is the relevant portion of my source material (emphasis added):
ICE-First-Aging is one of several experiments that investigates the effects of space flight on a model organism in the nematode worm family (Caenorhabditis elegans) and aims to develop links to human physiology in space. The organism chosen for this study is known to be able to mate, reproduce and develop apparently normally during space flight.
Fruit flies have also successfully reproduced in space, and there was an attempt to study the reproduction of geckos in space--unfortunately, the geckos froze at some point during the flight.
Till now no animals have been successfully reproduced in space.In 1979 Russians carried out an experiment in space to find out whether rats can reproduce in space , in that space mission the rats showed no signs of copulation because of weightlessness in space. Then Japanese researchers artificially fertilized mouse eggs with sperm that had been stored inside a three- dimensional clinostat, a machine that mimics weightlessness by rotating objects in such a way that the effects of gravity are spread in every direction. Some embryos were ultimately implanted in female mice and survived to a healthy birth, but at lower numbers than a regular-gravity control group because of the microgravity.
In answer to your second question three space experiments were carried out by the Russian scientists by using aquatic invertebrate animals such as Amphipods, Gastropods (pond snails), Ostracods and Daphnia (water flea) . The first experiment used a Space shuttle only and it was a 10-day flight. The other two space experiments were carried out in the Space station Mir (Shuttle/Mir mission), and the flight units had been kept in microgravity for 4 months. Daphnia produced their offspring during a 10-day Shuttle flight. In the first Mir experiment, no Daphnia were detected when recovered to the ground. However, they were alive in the second Mir experiment. Daphnia were the most fragile species among the invertebrate animals employed in the present experiments. All the animals, i.e., Amphipods, pond snails, Ostracods and Daphnia had survived for 4 months in space, i.e., they had produced their offspring or repeated their life-cycles under microgravity
Japanese rice fish, Oryzias latipes, spent 15 days in 1994 abord the Colombia Shuttle where they reproduced: http://cosmo.ric.u-tokyo.ac.jp/SPACEMEDAKA/IML2/e/text/textcontents_E.html
The same species where brought aboard the ISS in 2012, http://www.nasa.gov/mission_pages/station/research/news/aquatic.html, with facilities for breeding - though I havn't found the results of that experiment.
The first animal that conceived her offspring in space was a cockroach named Nadezhda. She was one of many cockroaches on board the Russian satellite Foton-M 3 (September 14 - 26, 2007) After they were returned to earth, the one named Nadezhda became the first earth creature to produce young that had been conceived in space. Her 33 kids were healthy though their carapace had darkened in colour much earlier, in comparison with natural-condition cockroaches who develop that darker tone later in their life cycle. But the rest of the conditions and capacities of the cockroaches remained normal.Later it was reported that Nadezhda's grandchildren, born to one of the space-born insects, had given birth on earth to normal cockroaches, with a life cycle and development pretty similar to that of any other cockroach. https://en.wikipedia.org/wiki/Nadezhda_(cockroach).
Here is a picture of Nadezhda and her 33 children.