This is just my opinion you don’t have to agree!!
“Sometimes you gotta bleed to know, that you’re alive and have a soul”
I think these lyrics are referring to pain in general, doesn’t have to be this situation, but say you love someone & are crying over them, the crying would be the “bleeding” whereas the reason why you’re crying is cause you care about them. When ppl say someone “has no soul” I usually think of someone completely emotionless. But by showing this emotion (crying) you’re expressing “that you’re alive and have a soul”.

“But it takes someone to come around, and show you how. She’s the tear in my heart.” To me this means that the pain from the first line is no longer general pain, but is now focusing on the pain of a break up or something. “It takes someone to come around, and show you how” I think the line “show you how” is someone showing him how to have a soul and be alive (which we know from before that sometimes you gotta bleed to have a soul). Meaning the ‘someone that came around and showed him how to have a soul’ was probably a girl who broke his heart and taught him pain but also taught him that he is capable of loving and being loved.
“She’s the tear in my heart” what I get from this, is that she was someone who broke his heart. No matter what, the ‘tear on his heart’ will always be there because she’s the reason it tore. That last part didn’t make much sense lol. The second possible meaning imo is this song is about the love of his life. He explains his attraction to her in the line “The songs on the radio are okay, but my taste in music is your face” and in the line “my heart is my armor, she’s the tear in my heart” I feel that means that although he is a tough guy, she is his one weakness because he loves her so much.