Does an electron produces magnetic field in an atom? I've been thinking that if a moving charge produces magnetic field then an electron moving around the nucleus of an atom must produce a magnetic field. And if it produces magnetic field then every matter in this universe must behave as a magnet or have some magnetic properties.
 A: Except for its intrinsic magnetic momentum related to its spin, an electron in an atomic orbital produces a magnetic field only if it possesses an orbital angular momentum, which also produces a magnetic moment. Note that the QM model of the atom also has orbitals without angular momentum, in contrast to the Bohr model. Thus , e.g., the lowest electron orbital of the hydrogen atom has no magnetic momentum and magnetic field. 
Therefore, an electron "moving around in an atom" does not necessarily produce a magnetic field.
Note: Ferromagnetism is a solid state phenomenon related to the spin of the electron.
A: In general, yes: the "motion" of electrons inside atoms (to the degree that quantum mechanics lets you speak of such a thing, which is very low) is associated with a magnetic dipole moment, which means that the material will have some degree of magnetic properties and interactions. However, in pretty much all materials, those interactions will be very weak and they are nowhere near comparable to the ferromagnetism that's produced by the alignment of macroscopic numbers of electronic spins.
OK, so, first off: for electronic motion inside an atom to contribute to its magnetic dipole moment, the first requirement is that the total orbital angular momentum of the atom needs to be nonzero. This rules out e.g. hydrogen and helium, whose electrons are in $s$ states that have no angular momentum, and it also rules out the atoms with full sub-shells (i.e. the noble gases, the alkaline earth metals, the right-hand end of the transition metals, and so on) where, roughly speaking, for every electron that's "going clockwise" about a given axis, there is another one that's "going counterclockwise".
The details of the magnetic properties of materials tend to be quite complicated, but the upshot is that if an atom has half-full shells, there's a reasonable chance that it will be paramagnetic, and it will be attracted by magnetic fields. Other materials are diamagnetic, and repelled by magnetic fields, a behaviour that can come from electronic spins but also from orbital motion, even in full-shell configurations, due to magnetic precession dynamics.
Generally speaking, though, those interactions will be very weak, and they do not match ferromagnetic materials, for which you need (i) a nonzero total electron spin in each atom, (ii) nearest-neighbour interactions that are arranged so that it is energetically favourable for neighbouring atoms to align their spins, and (iii) for the resulting magnetic domains to align in a macroscopically meaningful way. (See e.g. this video for more details.) For paramagnetic and diamagnetic materials, none of that holds, so the interaction is transitory (it doesn't stay if you remove the external field) and generally much weaker.
