3) Is it haram for my dad to refuse to bring tomatoes even though it is his job to clothe, shelter, feed us? I want to start cooking and I know it will require many tomatoes is it wrong for me to ask him to pay for that?
Allah knows best but I don't see why it would be your father's duty to provide you with every little thing you ask. A tomato is a specific item of food that if you never had, it wouldn't cause you to starve.
However, I would advise that you don't look at what is and what isn't his job and duty. Don't demand anything. If everything breaks down to demanding of rights and responsibilities and accusing others of not fulfilling their duties then your relationship will have no substance.
You can ask him politely to buy it and if he refuses, it's not the end of the world. You work and have money so you can buy it out of your own pocket.
That's just my advice.
4) Would him paying for laundry go in the "clothe" category? What about paying for cell phone, he refuses to own 1 and mine is 25 dollars pay as you go monthly. Would it be my job to pay for it since I use it mostly or his? I don't mind but I'm asking. He sometimes uses it. He refuses to get new mom 1, would that be her job or his?
Same as above for laundry. Ask him politely, if he refuses, pay it yourself - that's just the way it's gonna be. There is no need to feel bad about him for refusing.
The phone, if you use it mostly, it's only reasonable that you pay for it. Each time you lend it to him, consider it a service that you are doing for which you'll be rewarded inshaAllah.
5) My friend wants me to join a gym, but it is a coed gym. I know the answer but I would like another "opinion."
You know the answer but want people to give you incorrect opinions that you can use to justify joining such a gym?