Is a plasma necessarily made of monoatomic ions? Is it possible to have a plasma made of polyatomic ions instead of monoatomic ions? 
I want to know all the details why such a thing may be attainable or not and, if possible, what methods we can use to create such a substance.
 A: A plasma can be comprised of molecules so long as the temperature of the plasma is sufficient to ionize the molecule without dissociating it. Note that most definitions of plasmas do not really require a percentage of ionization (e.g., a gas that has only 1% of it ionized is considered a "cold" plasma whereas something much more than this is usually considered a "hot" plasma).
You can create a molecular plasma the same way as making a monatomic plasma: by heating the gas (obviously not to the point that the aforementioned disassociation occurs.)
A: First, your typical neutral plasmas contain both ions and electrons.  The ions may be polyatomic or monoatomic.  There also exist non neutral plasmas, which are composed of only ions or electrons.
Plasmas in the laboratory are typically (though not necessarily) created by first injecting a gas into a vacuum chamber and then first ionizing that gas using large electric fields.  The electric fields that initially break down the gas can come from different sources (e.g.,  toroidal electric fields in a tokamak, or large displacement fields in a capacitively-coupled antenna).  Once the gas has been broken down to some extent, the plasma is typically sustained by electron-impact ionization.  
Depending on the temperature of the electrons, and the composition of the initial gas (diatomic), one may either produce monoatomic or polyatomic ions.  If the electrons are hot enough, it will be difficult to sustain a polyatomic ion species.
A: Yes, a plasma contains positive ions and negative electrons, but the positive ions don't need to be atoms. Actually the negative charges don't even need to be electrons. For example, plasma etching of silcon is done using a sulphur hexafluoride plasma that consists of (mainly) SF$_5^+$ and F$^-$ ions.
I'm not sure that the word plasma has a precise definition, other than vaguely meaning a gaseous state containing charged particles but electrically neutral overall.
A: Plasma  is essentially ANY ionized gas that has been electrified with extra electrons in both negative and positive states.  If you were able to create an Oxygen plasma with polyatomic ions, it would still be a plasma.  I would try to mix a second gas with Oxygen if I was attempting to do this.  Here is my source: Plasma Treatment Basics
