Sounds like you are a normal teenager. Perhaps a little sensitive but that is a very good thing. You see the world is full of macho masculine men (esp. in the United States). And there are a shortage of people.... men who are emotionally mature which has nothing to do with physical maturity or strength. In fact, often there is an inverse relationship that is to say that people who focus on being manly end up emotionally retarded.
It sounds like what your father wants is for strength of character. The military can help the right person but it sounds like you are not that type of person. I would revel in this. The military is mostly for people who are good at being programmed to follow orders, work as a unit. The people I know in the military in the US are hard working generally good folk but are not that interesting or dynamic. They are varying degrees of simple minded, many of them.
If your father is judging you based on an idea of masculinity that is focused on being 'thick skinned' tough and unemotional than that is no better than being too sensitive.
I will tell you that their is two main types of intelligence; IQ and EQ. The second means emotional intelligence. These are people that understand what other people mean and are feeling. You are young still but the type of person you are describing yourself as sounds like a person that will grow up to have emotional intelligence. You will overcome your adolescent fears and your sensitivity will become a great strength.
I was sensitive too as a youth. Now at age 33 I use that sensitivity to form relationships with other kind caring people. The same sensitivity that caused be to over react and be afraid of knowing people when I was a teenager is the sensitivity I use now to make other people feel that they can trust me. This is more important than anything else because people who can trust you will often help you out. They will be your friend because they feel you are a friend to them.
This extends to business relationships. Today I'm filling out forms to buy an investment property that will add money to my income every year. I was sensitive to the seller of the property and he was patient while I worked out the financing. I was sensitive to the bankers and they helped me get a loan. I am sensitive to the people who rent it out and they like me and will probably come back again and again. So being sensitive can pay!
Anyone who thinks you should be more manly is ignorant of the fact that their is no real man or real woman. Just real people. The idea of a 'real man' is ridiculous because it is prejudice and seeks to limit the possibilities of what character you can form into a small little box. It is over simplifying and stupid.
Think outside this box young man! You are smart and wise to listen to artists and poets. to writers and scientists. The world is becoming more welcome to people who think and feel and those masculine men your parents want you to become will be left being the grunt laborers. And you will be their boss!!
Although to the point about overcoming your fears that is okay. Take little steps. If you need to move a dead rat, try being in the room and just looking at it. Take deep breaths. and calm your self. Next try using a shovel to move it... Know that when people make jokes they are trying to get a reaction from you. Usually people just want you to think they are funny even when they tease you they are more trying to get affirmation. I would laugh at their jokes. That makes it about the joke not about you. Also confident adults will make jokes about themselves that they tell others. It's called self deprecating humor. You might try this out.
It sounds like you are sad about your grand parent dying and it sounds like your father is realizing his own mortality through his moms dying. It's hard for them. They are just people like you and I, sensitive on the inside no matter how tough they try to be. So try to be understanding of them and listen to them and most importantly ask them questions and remain calm even when they are upset if you can. This will command respect after a while. Be confident in yourself because you care about others you are a good person and someone that people like me highly respect.
Any more questions feel free to write;
genegenre@gmail.com