Do plasmas have their own magnetic fields? Do Plasma generate a magnetic field of their own ? Can they cause electromagnetic induction?? 
For example if a hydrogen plasma is ionized and its electrons removed.. And the protons accelerated through coils(With the help of electromagnets) would it generate electricity in these coils? 
If they have magnetic fields how is its strength measured
 A: Background
Electric and magnetic fields are produced by sources, e.g., charge and current, respectively.  The particle electric current density found in Ampere's law is defined by the following equation (already in macroscopic form$^{\mathbf{A}}$):
$$
\mathbf{j} = \sum_{s} \ q_{s} \ n_{s} \ \mathbf{v}_{s} \tag{1}
$$
where $\mathbf{j}$ is the 3-vector electric current density, $q_{s}$ charge of particle species $s$, $n_{s}$ number density of particle species $s$ (i.e., number per unit volume), and $\mathbf{v}_{s}$ bulk flow velocity of particle species $s$.  Thus, if there is a relative drift between two oppositely charged particle species, one can have a net current density.
In nature, plasmas are satisfy what is known as quasi-neutrality whereby one can consider a plasma a neutral gas over length scales larger than the Debye length.  It is possible to generate a charged plasma in a lab setting using a large potential grid to separate ions from electrons, but in nature the electric fields produced by such a charge separation would act to eliminate the separation immediately.  Since natural plasmas do not have the benefit of external "mechanical forces" or "batteries" like in a lab setting, it is typically a safe assumption to consider them a neutral gas.

Do Plasma generate a magnetic field of their own?

Yes they can if there are currents (i.e., Equation 1) in the observation frame of reference or a time-varying electric field (i.e., the displacement current).  Another mechanism that can generate electromagnetic fields is called a plasma instability, whereby some source of free energy (e.g., a beam in an otherwise homogeneous system) drives the system unstable.  The system can respond (i.e., the instability) by radiating an electromagnetic wave, which converts the free energy into electromagnetic energy.

Can they cause electromagnetic induction?

Yes, the plasma emitted by the sun induces electromagnetic fields in Earth's upper atmosphere and in power lines (e.g., see answers here and here for further discussion).  There is a legitimate concern that some solar storms may cause power grid blackouts as well, discussed in detail here.

If they have magnetic fields how is its strength measured

There are two standard instruments used in space plasmas called fluxgate and search coil magnetometers.  A fluxgate magnetometer takes advantage of the hysteresis curve of a ferromagnetic material by measuring how much more or less applied field is required to reach the saturation levels.  A search coil magnetometer takes advantage of Faraday's law to measure the induced fields within a coil of wire wrapped around a core.  The difference between the two is that a fluxgate can measure DC fields (i.e., quasi-static) up to a frequency well below the driving frequency (i.e., rate at which the applied field cycles between saturation levels on the hysteresis curve) while search coils measure AC fields (i.e., time-varying) and their range of frequencies depend upon how they are designed$^{\mathbf{B}}$.
Side Notes
A.  To see more details on the difference between micro- and macroscopic Maxwell's equations, see pages 248-258 in Jackson [1999].
B.  The details for designing a search coil magnetometer are rather messy as it is a resonant circuit and is limited by multiple factors including the length of the core (influences range of frequencies), number of windings per unit length, impurities in the wire (e.g., affects noise levels), electronics (e.g., affects noise levels and limits frequency range), etc.
References


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*J.D. Jackson, Classical Electrodynamics, Third Edition, John Wiley & Sons, Inc., New York, NY, 1999.

