How would I push a negative number with OPCODES onto the stack? If the user wants to push -10 onto the stack.

Would I do "10 OP_NEGATE" ?

Or could I convert it to the hex value by doing 0x80 + 10 = 0X8A ?
Both will work. OP_NEGATE simply XORs with 0x80, so you can also directly push the result.