Even if the atom is neutral (equal numbers of protons and electrons), the electrons and nucleus form an electrical dipole, so there is still an electric field around them, even though the total charge is zero. Is this reasoning correct? So, every atom has an electric field, even if it is too weak? If not, why?
In an atom the electrons do not have a position. They are delocalised over the whole atom and in the absence of any external field this results in a symmetric charge distribution. So an isolated neutral atom is spherical and has no external field.
However if you bring two atoms near each other then their charge densities will develop fluctuations that are correlated. That is, both atoms will develop small but non-zero fluctuating dipoles and indeed higher multipoles. The interaction between these is the origin of the London dispersion force. So in these circumstances the atoms do have a fluctuating external field, though the long time average still remains zero.
Strictly speaking, yes, I believe neutral atoms have an electric field (though very weak). For example, if you have a hydrogen atom in the ground state, the absolute magnitude of the wave function of the electron decreases exponentially with radius (for a sufficiently large radius), so there is a (very small) positive total charge within a sphere with an arbitrarily large radius.
Atoms are in the quantum mechanical regime. The electrons around the nucleus occupy orbitals ( not orbits) i.e. probability loci where a measurement will find an electron.
These orbitals have quantum numbers that give them shape. When the angular momentum quantum number l is equal to zero, the orbital is isotropic and neutrality is maintained. But for higher l values there are shapes, as seen in this table. Thus the negative charge has a shape, that leaves space for the positive charge of the nucleus to come out, creating positive and negative fields. These are the fields that generate the bonds for molecules, solids and liquids.
Edit after comments discussion
It is not simple, because this is a quantum mechanical frame.
Here is a measured orbital distribution for hydrogen , and it is symmetric.
The difference with the images in the table linked above is due to the fact that the experiment shows a time averaged distribution of the probable locations of the single electron around the proton for many randomly oriented hydrogen atoms. At any given time t, the electron will be at a single point with respect to the proton, and a field shape will appear. When two hydrogen atoms approach each other, the proton is not completely shielded by the electron, but an effective potential between the two atoms can be established leading to the bond that makes shared orbitals for the two electrons and creating the stable H2.
So yes, there is an instantaneous spill over field, whose probable strength that depends on the probabilities given by the orbitals .