Do you have to feel to show empathy?

I hate when people automatically thinks you're a bad person when you lack empathy. Like, no? I may not react the same way everyone does on certain situations, just because you found it sad doesn't mean I will either. Even though I lack empathy my sympathy is quite high. I may not feel sad or show any reaction at someone's death, but it doesn't mean I'm not gonna comfort their loved ones and offer support. Yeah I don't feel sad but I understand why they're sad and it's enough, so I'm not a bad person...I hope. 

Still, there're sometimes when I feel bad for not feeling anything. For example, when someone tells me about a bad experience they've had gone through(sexual assault ect) I don't feel sad (even if it's my friend) like I genuinely can't feel it inside me, but I'm mentally sad? If that makes sense. I feel sorry for them and what they've been gone through, but sometimes I feel like I'm tricking myself and I actually don't care, since I can't feel it. 

Do you have to feel to show empathy?

Would you find it weird if one person was just keep doing their own thing and staring blankly to everyone while everyone in the room cries for the said person's family member? Are they the odd one? 

Do you have to mask every reaction to seem like a good person? 

I think I'm good. Me being good doesn't have to do anything with my reactions or feelings. Still I can't shake the thoughts.