Do I have an "anti-research" personality?

1. Having a broad set of interests and knowledge is fine, and can lead to some 'outside of the box' solutions. Yes, you will find periods where anything else is more interesting than your PhD work, irrespective of how interesting your PhD work actually is. You'll find that as you come closer to the end of your PhD, your concentration on the problem at hand will sharpen, and while a PhD deadline isn't quite as good at sharpening a mind as an impending duel, you probably will find your ability to concentrate improve as the ending comes into sight.

   Furthermore, don't assume what you're researching your PhD on is what you'll be doing for the rest of your life. Having a wide set of skills and a shallow knowledge of many things will often help you out when you (most likely) realise you don't have the desire to become an academic, and need to search for a different job (which, let's face it, given your background means probably going in to Data Science)

2. I don't know too much about the culture of applied math to make an informed comment on this, but in general, finding a good real-world application of a concept, and carrying it out, is difficult. It can involve a lot of time mucking around with real world datasets, which are often messy. You have an obligation to show that your results and methods are useful, but to what degree is up to taste, and I don't think is an issue with any kind of 'anti-research' personality.

3. This sort of thinking is by far your most serious problem. If asking fellow students for help and collaborating is cheating, then so is using Stackexchange or reading from a textbook. Modern civilisation rests on the ability to record our thoughts so every human doesn't have to derive all the rules from scratch. There is nothing dishonourable about asking for help. If someone gives you helpful ideas, thank them in your thesis. Even if they do clever novel work on your problem, you still have to at the very least understand it and synthesize it into your own work.

   There is a certain kind of person who is reluctant to ask questions. I would count myself as one. If you feel uncomfortable doing do, I recommend working on that skill until you do feel more comfortable. Stackexchange is the perfect place for that - even if it isn't a question about anything to do with your research.

4. This behaviour is fine. One can get a certain fatigue when working on certain projects for a while.

5. & 6. It does really seem like you have a need to derive and understand everything from its foundations. In real life, this is rarely necessary.

   I did a PhD in organic chemistry. After that, I decided I didn't like chemistry anymore and went on to be a Data Scientist. At first, I would spend a lot of time trying to mathematically understand the techniques I was using, and I wouldn't feel comfortable using them if I didn't understand the underlying linear algebra. Then I slowly got more comfortable using techniques whose underlying mechanics I don't understand. We by and large don't understand the inner workings of a car or computer. We do know, roughly, its outputs when given certain inputs. The inner workings are a problem already solved by someone else.

   As you grow older and read more, you'll be continuously struck by the sheer quantity of stuff out there that people have figured out. The sheer amount of knowledge that goes into, say, building an aeroplane or a computer game or a treatment plan for a specific disease or a skyscraper or a hadron collider. Letting go of the need of having to understand all aspects of a certain thing is an important skill. Practice triage with the information you receive and the problems you encounter. Trust in the expertise of others who have come before you. Know when to expend your brainpower, and know when to conserve it.

So in summary, apart from issue 3, I don't think you have any significant "anti-research' personality issues.

Let me preface my answer by challenging an unwritten assumption of yours: that you have a fixed "personality" which is hampering your potential as a researcher. This simply is false. Nobody is born a good researcher. It takes hard work to become one. Sure, some people have predispositions that help them. But in the end, you cannot avoid doing the work, including work on yourself.

You have discovered that you lack certain skills, or that your aptitude for them is currently insufficient. Blaming this on your personality as something that you cannot change is lazy. It is up to you to improve in these areas. If you refuse, then you are to blame, not some god-given "personality" that was forced upon you.

I will now answer point by point, but if you are honest with yourself, you already know what I am going to say.

1. There is a common saying: "jack of all trades, master of none". In math, you are studying concepts that other people have been studying and refining for decades if not centuries. You are not going to master them by studying them for a few months and moving on to the next target. You are now at the bleeding edge of research, and it takes hard work to get up to speed. Your PhD is the occasion that you get to demonstrate that you are able to become a true expert at something and make improvements to the state of the art. It is unlikely that you will manage that by working on something for six months and moving on to something else afterwards.
   However, getting interested in other things is not just fine, it is mandatory if you want to have a successful career after your PhD.
2. You seem to be confused about what applied math really is. Math that you actually apply in the real world is usually called engineering. It is a one-in-a-million occurrence for something in applied math to find a real-world application. But since you don't know in advance what will work, you have to start somewhere. Toy models are where it's at. You start with something simple. Maybe it works. Maybe you realize it's more complicated than what it appeared (the most common occurrence). Maybe it turns out to be useless. You don't know, because you are not an oracle.
3. There is nothing "fake" about getting influenced by others. Here is another common saying: in academia, we are standing on the shoulders of giants. You are reading papers, aren't you? You didn't discovered all of applied math by yourself from scratch, did you? Aren't these papers influencing your research, your way of thinking? Have you never read something and thought, "hey, I never thought about it like that", or "wow, this unrelated topic gave me ideas for my current research"? The difference with face to face conversations is that it's happening in real time. That's basically it.
   And here's another saying, as I like them: alone you go faster, together you go farther. Other people have a different way of thinking, a different skillset, a different knowledge of the literature, different ideas. By pooling your resources together, you will probably be able to solve bigger problems than if you were on your own.
   Finally, collaborating is just fun. Try it and you will see.
4. See #1. If you are happy with making tiny contributions to the state of the art all the time, good for you. (This is especially at odds with your apparent disgust for people who "pump out" papers, but whatever.) Meanwhile, someone will study seahorses for ten years and revolutionize the field by realizing that seahorses are aquatic poneys, rendering most of your previous research mostly useless.
5. Solving things on your own is fine and valuable. Being able to use other people's ideas to solve your problem more quickly is an extremely valuable skill, however. While you will be reinventing the wheel, others will go around the world in a plane.
6. This is something between you and your advisor. As a PhD student, you are supposed to transition from supervised research to autonomous research. If you really feel this way, talk about it openly with your supervisor. Maybe you will be surprised. Maybe your supervisor even thinks that you are not ready for unsupervised research because of your other points (unability to focus, to collaborate, to make use of other people's ideas and applying them to your own problem…) and this is why he is still asking you to work on something where he can directly help you and direct you in your research, instead of letting you work on something with less direction and help.

So yes, if you continue with the mindset that there are certain things about yourself that you cannot improve due to your "personality", you will have a hard time, in academia and elsewhere. You will continue to give up at the first obstacle and never amount to much. And you can continue to think that nature, your personality, your upbringing… is to the culprit, in the end, you are the one who has to live with the consequences.

Short answer to complement other longer good ones here.

You do seem really to care about the seahorses of all colors, and the pipefish and the shrimp. So focus now on learning whatever you can in your current environment. Continue reading stuff that's not directly applicable. Do collaborate. Finish your PhD (toy models are sometimes good places to start). Then look for a postdoc with someone whose philosophy and style better match yours. When you run your own lab, look for students like you.