Then does all charges, no matter how much charged are they, ionize the material around them since distance can get smaller and therefore electric field gets bigger. Consequently, the electric field exerted by the charge would be higher than the electrical breakdown limit of the material around it.
Let's say you magically stick a (positive) charge inside a piece of material. You can imagine getting as close to a charge as you want, but the "material around it" is still made up of atoms which are a certain distance away. The atoms of the material are not a continuum which gets infinitesimally close to your charge. The electrons of the material may get close, but you have to think of the electrons quantum mechanically: they are "smeared out" in space, and to avoid getting into that here, the small region where the electric field is arbitrarily high gets averaged out from the point of view of the smeared electron.
2-My other question is more basic. When we try to calculate the electric field between a capacitor which has two oppositely charged object, object charged with negative sign would exert a negative electric field whereas object charged with positive sign would eert positive electric field. If the amount of charges were the same in each object, would the positive electric field and negative electric field cancels out each other and would the net electric field be 0? I know it doesn't work like that but I want to get the correct information.
You have to consider not just the magnitude, but also the direction of the electric field. Say the positive charge is above and the negative charge is below. The positive charge creates an E-field pointing away from itself, so, inside the capacitor, that points downward. The negative charge creates an E-field pointing towards itself, so, inside the capacitor, that points downward. Thus both E-fields actually add together to make a stronger one. If you play the same game but outside the capacitor, you'll find the E-fields cancel each other (all of this assuming you do the standard infinite capacitor plate).
Your question touches the question of ontology in particle physics. Historically we are used to be thinking of particles as tiny independent entities that behave according to some laws of motion. This stems from the atomistic theory of matter, which was developed some two thousand years ago from the starting point of what would happen if we could split matter in ever smaller parts. The old Greeks came to the conclusion that there had to be a limit to that splitting, hence the atom hypothesis was born.
This was just a philosophical idea, of course, until around the beginning of the 19th century we learned to do chemistry so well that it became obvious that the smallest chunks that matter can be split into seemed to be the atoms of the periodic table. A hundred years later we realized that atoms can be split even further into nuclei and electrons. What didn't change was this idea that each chunk had its own independent existence.
This idea ran into a deep crisis during the early 20th century when we discovered the first effects of quantum mechanics. It turns out that atoms and nuclei and electrons do not, at all, behave like really small pieces of ordinary matter. Instead, they are behaving radically different, so different, indeed, that the human imagination has a hard time keeping up with their dynamic properties.
For a while we were in a limbo regarding our description of nature at the microscopic scale. It seemed like we could cling to some sort of "little weird billiard ball with mass, charge, spin etc. properties" kind of theory for electrons, but as time went by, this became ever more hopeless. Eventually we discovered quantum field theory, which does away with the particle description completely, and with that all the ontological problems of the past century have disappeared.
So what's the new way of describing nature? It is a field description, which assumes that the universe is permeated by ONE quantum field (you can split it up into multiple components, if you like). This quantum field has local properties that are described by quantum numbers like charge. This one quantum field is subject to a quantum mechanical equation of motion which assures that some properties like charge, spin, angular momentum etc. can only be changing in integer (or half integer) quantities (in case of charge it's actually in quantities of 1/3 and 2/3 but that's a historical artifact). Moreover, this field obeys symmetry rules that leave the total sum of some of these quantities unchanged or nearly unchanged. Charges in particular can only be created on this field in pairs such that the total charge remains zero.
So now we can answer your question in the language of the quantum field: the electron gets its charge by the field allowing to create one positive charge state and one negative charge state at the same time, leaving its total charge zero. This process takes some energy, in case of the electron-positron pair a little over 1MeV. Every other property that is needed to uniquely characterize an electron is created in a similar way and at the same time. The elementary particle zoo is therefore nothing but the list of possible combinations of quantum numbers of the quantum field. If it's not on the list, nature won't make it (at least not in form of a real particle state). Our list is, of course, at best partial. There are plenty of reasons to believe that there are combinations of quantum numbers out there that we have not observed, yet, but which are still allowed.
Best Answer
It is most definitely the magnitude of charge that matters, reversing the charge all it does is make a mirror image circle, so since in your experiment, you are measuring inherently sign-less/directionless parameters like radius, time, and magnetic field, speed etc. You would get a positive value for e/m. However, when you do report the value of $e/m$, you must report it with a sign, i.e negative. So, the ratio should be negative but, from the experiment As we are dealing with directionless quantities, you get a positive number as an output for the experiment, but when someone asks for the specific charge, not the magnitude then you report it with a sign.