Why is it 'syn' and not 'synem'? Does the fact that 'to' is present change the casing, even though 'jest' normally requires the instrumental form?
Yes. "To" before "jest" changes the cases from instrumental to nominative. It is also the word that does not require "jest" so "to mój syn" is also correct.
https://www.duolingo.com/skill/pl/Defining- read the part about "to"
Why can't i use the word Ten?
"To jest" is fixed, "to" doesn't change with gender in this construction.
Why does it it use the 'mojego' pronoun here?
Sorry, what's the question? There's no 'mojego' here, that's a very simple sentence...