What material fills the synaptic cleft? Is it water?
There are two types of synapses namely Electrical synapse and Chemical synapse. In electrical synapse there is physical contact between two cells through gap junctions. In chemical synapse there is a small gap between two cells which is termed as synaptic cleft.

The presynaptic and postsynaptic membranes at chemical synapses are separated by a synaptic cleft that is 20–50 nm wide, 10 times the width of the separation at gap junctions. The cleft is filled with a matrix of fibrous extracellular protein. One function of this matrix is to serve as a “glue” that binds the pre- and postsynaptic membranes together.(reference 1)

The same reference also states Water is the main ingredient of both the fluid inside the neuron, the intracellular fluid or cytosol, and the outside fluid that bathes the neuron, the extracellular fluid. Electrically charged atoms, ions that are dissolved in this water are responsible for the resting and action potentials.

Reference 2 mentions, The synaptic cleft is not simply an empty space, but is the site of extracellular proteins that influence the diffusion, binding, and degradation of the molecules, including neurotransmitters and other factors, secreted by the presynaptic terminal 

If we think in this direction, of course holding together two neurons at synapse position is important. Moreover, the presence of several dissolved ions (in water) are necessary. If you remember, for the release of neurotransmitters into the synaptic cleft, influx of Ca2+ ions through voltage gated ion channels is essential.Thus, from above all points we can conclude that, the extracellular fluid in synaptic cleft is mainly aqueous in its composition with several ions, released neurotransmitters and several proteins that constitute a matrix.

What force makes the neurotransmitters move toward the receptors?

An essential role of astrocytes is regulating the chemical content of this extracellular space. For example, astrocytes envelop synaptic junctions in the brain, thereby restricting the spread of neurotransmitter molecules that have been released.(reference 1).Thus, it helps to confine the neurotransmitters in the small region, thereby increasing the probability of getting in touch with the target receptor.Also, as reference 2 says, there are several proteins present in the synaptic cleft which influence the diffusion.It means movement of neurotransmitter is by diffusion and above mentioned various factors can influence their movement towards receptor efficiently.

References:


Bear et.al, Neuroscience, exploring the brain, 2015
Neuroscience, textbook by Dale Purves.