Warning Signs of Abusive Relationships?

I've only been with this guy for a month and a half. We moved really fast. Quickly fell hard for each other. A ton of sexual attraction. He showers me with flowers and dates, holds the door open, makes me feel so beautiful and loved and appreciated. I thought, "too good to be true," wouldn't anyone? He even has said that he already wants to marry me. Lately, I started noticing some of the things he does, and I don't know if I'm paranoid...? 
1) He is in the military and talks about how he can't wait to go and kill people?
2) He has severe anxiety and depression, so he is constantly worried about what I am doing to the point that it's almost crazy. He texts my best friend every other day, asking if I'm okay if I don't respond within minutes or with a bunch of happy emojis. I thought this was cute.. Now, it's extremely obsessive.
3) He has willingly told me that he "needs" me. I've tried to clarify this, to ask if he was actually serious, and he is. He's even said he's obsessed with me. He carries a picture of me everywhere with him.. Cute, right? He puts it in his uniform. He also tapes it on his wall before bed or even sleeps with it. 
4) He stares at me. Always. He just stares and stares and tells me how beautiful I am, and he's super over protective of me when it comes to other people messing with me. 
5) After our first fight.. And our only fight... He grabbed my covers off of my bed, threw them, grabbed both of my arms and pulled me up forcefully and demanded for me to look at him while I talked to him. He was literally begging me. I kept saying no, I wanted space, he needed to leave me alone, but he wouldn't take that for an answer and kept pushing my hands away from my face since I was trying to cover my face. 
6) also, with our fight, it was over him lying about something and I caught him lying. He immediately apologized and became really stern and that's when he burst into my bedroom and threw my covers off wanting to talk about it. He ended up CRYING over it the next day, put the blame on someone else, he was trying to make his lie seem a lot less worse than it was, and was acting as if j should be ashamed for letting him feel horrible after what he did. Almost as if he wants to do a 360 and put this on me.  
I am not used to this.. I don't know if these are signs of a possible to-be emotional or physical abusive relationship or if he's just really passionate and has a little temper sometimes. I just don't know. Like I said, I'm not used to being treated like this so I don't know if I'm just used to being treated like shit  or if this is weird.