In the first question, if the instruction NVIC$\_$Enable$\_$IRQ$0$ is removed: a$)$ False, the processor won't operate infinitely in sleep mode. b$)$ False, the PORTD interrupt activation isn't directly affected. c$)$ True, no interrupts will occur, potentially leading to the microcontroller operating infinitely in sleep mode. d$)$ False, as option c is true. In the second question, the modified PORTD$\_$IRQHandler function alternately increments the "count" variable by $4$ and $6$ each time a high to low transition occurs on PTD$6.$ This ensures the alternation between the two increment values, maintaining count's variable behavior.