Am I hearing correctly that the j at the beginning of "jsem" is silent and only written for fun?
Yes, it is generally dropped. (But it is pronounced if there's a prefix, like in "nejsem".)
Some speakers will also often pronounce it, especially when trying to sound formal.
why not same as Matěj?(Já jsem Matěj)
Apparently you can use the pronoun or not. I know Spanish is like that: "Yo soy NRPL" or "Soy NRPL" are both correct. Italian is like that, too.
A lot of languages are pro drop, such as Spanish, Italian, Romanian, Portugese, Greek, Polish, Hungarian and many more that I can't think of
Really getting into this. What does SG. Stand for?
Singular. (Most likely. Depends where you saw it.)