Le us start with what is a field in physics :
A field is a physical quantity that has a value for each point in space and time.1 For example, in a weather forecast, the wind velocity is described by assigning a vector to each point on a map. Each vector represents the speed and direction of the movement of air at that point.
A field can be classified as a scalar field, a vector field, a spinor field or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or a tensor, respectively. For example, the Newtonian gravitational field is a vector field: specifying its value at a point in spacetime requires three numbers, the components of the gravitational field vector at that point. Moreover, within each category (scalar, vector, tensor), a field can be either a classical field or a quantum field, depending on whether it is characterized by numbers or quantum operators respectively.
Secondly, the concept "electromagnetic" is stretched , there are electric fields and magnetic fields in classical electrodynamics, which combined create the classical electromagnetic wave. The electromagnetic wave is composed in a functional way by electric and magnetic fields but is not a field per se.
When we are talking of creation and annihilation operators, then we are at the quantized frame of second quantization. The field in this case is an operator defined over all space time points which operating on the state function will generate a measurable value . The classical electromagnetic wave emerges from a confluence of photons, and analogously the classical electric field and the classical magnetic field will emerge from the virtual photons in a mathematical confluence .
If the electromagnetic field doesn' exist, what exactly is the Higgs field? What makes the EM field different from the Higgs field?
There is no electromagnetic field because of the way the data have been modeled with Maxwell's equations. There exist electric and magnetic fields, as explained above . The Higgs field is an expression of the electroweak sector, and it is a field for physics, according to the definition of field above.
If the Higgs field exists, is it uniform? Does it have the same value everywhere in the universe?
Yes, it is uniform because it is a quantum mechanical field, an operator as defined above. As we do not have a theory of everything, we cannot really talk of the universe. In the energies we live in it has the vacuum expectation value, vev of 246 GeV.
The discovery of the Higgs boson at LHC supports the existence of the Higgs field and the symmetry breaking mechanism of the standard model.