Are all moons in our solar system rotating synchronously around their planet? If not, what are the criteria that some do and some don't?
Many thanks for any insights!
You are asking about tidal locking.
Some moons are tidally locked, others are not. There are several factors that lead to or militate against tidal locking:
There are other factors, and there are exceptions. One example is Hyperion, a 270 km wide moon of Saturn, which orbits Saturn at less than half the distance as does the much larger Iapetus. Yet Hyperion is not tidally locked while Iapetus is. One reason is size and shape. Iapetus is fairly large and is thus more or less spherical. Hyperion is roughly half of the potato radius. Potato shaped objects have funky rotations. Another factor is that Titan, Saturn's largest moon, has a strong influence on Hyperion orbit and rotation. Hyperion's rotation is chaotic.
Most of the irregular moons do not quite have synchronous rotation:
Regular satellites are usually tidally locked (that is, their orbit is synchronous with their rotation so that they only show one face toward their parent planet). In contrast, tidal forces on the irregular satellites are negligible given their distance from the planet, and rotation periods in the range of only ten hours have been measured for the biggest moons Himalia, Phoebe, Sycorax, and Nereid (to compare with their orbital periods of hundreds of days). Such rotation rates are in the same range that is typical for asteroids.
Some of them even rotate chaotically. Examples: Hyperion (which rotates so unpredictably that the Cassini probe could not be reliably scheduled to pass by unexplored regions), and Pluto's Nix, Hydra, and possibly Styx and Kerberos, and also Neptune's Nereid.
You can find more information in the following posts:
At Pluto the non-locking condition becomes quite notable. Pluto and Charon are double-locked, of course, but the smaller moons that orbit farther out all rotate much faster than their orbital periods. In the most extreme case, Hydra rotates about 90 times about its axis for every orbit it makes around the Pluto-Charon doublet. See the Infobx at https://en.wikipedia.org/wiki/Hydra_(moon).