It is not possible to levitate anything that doesn’t incorporate a material that can be magnetized, called a ferromagnetic material.

However, some creatures do incorporate magnetic materials. The scaly-foot snail lives in extreme conditions: hydrothermal vents at the bottom of the ocean almost two miles deep, where water heated by magma below the earth’s crust enters the ocean. This snail thrives in these conditions, and develops a shell that incorporates iron-containing materials, like a suit of armor. So, if you could find a really big magnet, you could theoretically levitate the snail!

Theoretically, yes - water is repelled by magnetic fields, and it is possible to move something containing water away from it.This would require such an incredibly powerful magnetic field that short of a magnetic neutron star, you're never going to have one.

This is a very interesting question. It is supposedly possible to magnetically levitate a human, or other organisms/objects without the assistance of metal objects. It has been suggested that using the fact that water is a diamagnetic material (materials which are repelled by magnetic fields), and that the human body is comprised of mostly water, a person could be levitated by applying a very strong magnetic field to him or her. However, it has been suggested that the magnetic field would have to be incredibly strong -- the order of several 10s of Teslas (a unit which described the amount of magnetic flux density an object has).

To put this into perspective, a strong refrigerator magnet would exert about 0.01 Tesla; this means that it would take something thousands of times stronger than a refrigerator magnet to levitate a person.

If you are interested, an estimation for how much magnetic flux density would be required to levitate a person can be calculated using the following information: according to Wikipedia, the minimum criterion for diamagnetic levitation is:

B*dB/dz = mu_0*rho*g/chi
where B is the magnetic field
dB/dz is the rate of change of the magnetic field in the vertical direction, z
mu_0 is the permeability of free space
g is the acceleration due to gravity
rho is the density of the material
chi is the magnetic susceptibility of the material

Yes, actually frog levitated by magnets is well-known in the physics community. All materials have weak diamagnetic properties (repels against the applied magnetic field direction) to a certain degree and most of them are overcame by other magnetic properties (such as ferromagnetic or paramagnetic). In some materials, the weak diamagnetic is the dominating magnetic property and good for the levitation experiment. Water is a good example.

A living animal such as a frog is good for the demonstration of magnetic levitation, because of the high percentage of water inside a frog. When you applied the strong magnetic field (at least 10Tesla), a frog can be levitated. There are three points that could make the levitation easier to observe:

1) the stronger and dominating diamagnetic the better,
2) the stronger applied magnetic field, the better;
3) the smaller gravity, the better.

Usually the first point is quite different comparing different substance, so the difficulty to levitate different materials varies.

Here is the video demonstrating such effect:
video