Blood being a necessity in virginal sex is a fairly common misconception. Blood comes from the hymen tearing and it's not supposed to tear, as it's not a solid wall. It's supposed to stretch. It can be stretched enough due to various forms of physical activity or the hymen could just be naturally more elastic. It doesn't seem like Hisao went incredibly fast in Hanako's sex scene, so he likely wouldn't have torn her hymen. Shizune, Rin, Lilly, and maybe Emi are all virgins as well, yet none of them bleed during the sex scenes.
That logic is heavily flawed, even for people with depression/extreme anxiety/whatever else you'd like to put here. While it may be difficult to understand the whole situation for them, a lonely person wouldn't just throw away a friendship or relationship because they feel lonely. The worse their mood gets, the less logical their thinking would probably be, but that's still too far. It's not hard to see that if you feel lonely and you just throw away a fairly large part of the relations (friendships and relationships) you have, then you're going to feel even more lonely. Hanako stopped believing in friendship after the accident (the house fire) when people recoiled at the sight of her, teased her relentlessly, and many of the adults at the orphanage didn't want to bother with her either. The worst part was that the people she thought were her friends were some of the ones calling her names and bullying her. However, through Hisao and Lilly's caring actions and words, she tried to, but couldn't, believe that friendships and relationships weren't suitable for her. Both Hisao and Lilly meant so much to her that they were the proof she needed to know that friendships and relationships were still a possibility for her. Lastly, Hanako is afraid of being abandoned by those whom are the closest to her, it's happened before with her 'friends.' Hisao thought Hanako was rejecting him after her birthday. She quite clearly tells him that wasn't the case at all.