Do I have an "anti-research" personality? I am a second year Ph.D. student in a North American university who is mostly working on applied mathematics with a supervisor/advisor/PI. I have been troubled by a pattern of my own behaviour that seems to go against the seemingly correct way to succeed in research and I wonder if I have some sort of an "anti-research" personality.

Here are the problems:


I study whatever is interesting and cool (within a reasonable range of my discipline) and this causes me to lack a solid focus. It seems to me that people who wind up obtaining a PhD learn to abandon so-called academic freedom early on and drill into a tiny research area. For example, while other people's research could be about different types of seahorses, my research would be about seahorses, pipe fish, shrimps...you get the picture. 
Since I work in applied math, therefore I often demand more proof of applicability, where my supervisor is totally happy with toy models. He thinks it is completely sufficient for publishing a paper while I think it is fake applied research. This seems to generate some conflict between us and usually I am the one who gives in at the end and do some simulation on these toy models to make him happy - I never care about these problems. Most other people in my field only simulate toy models as well, not really "real-world" applicable in any sense, but hey, it is sure pumping out a lot of papers.
I never collaborate or ask for help from my peers no matter how stuck I become. I often see my colleagues collaborate with each other in the office. A part of me is jealous that they are taking advantage of each other's knowledge, another part of me almost thinks that this is like cheating. I feel that my thesis and my ideas wouldn't be mine if it was interfered or influenced this way. It wouldn't feel authentic
I move on to a completely new topic immediately after I finish my old one. Usually the way I see how other people work is that they try to do something tangentially similar to what they have done before, i.e., blue-tailed seahorses, then yellow-tailed seahorses. I jump to studying pipe fish immediately after I warp up my research on seahorses. I think my advisor must dislike me immensely for starting from scratch after every project. 
If a problem seems to have an easy or existing approach that solves it, I find the problem less attractive to the point that I get physically sick from reading about other people's approach. However, a lot of times in research I find that other people would simply adapt other people's method for their own problem and then publish a paper that way. I try to solve it in my own way every time and most of the times it gets to a point where my supervisor intervenes and tell me to do whatever other people are doing. Again, I give in at the end.
I don't even try to follow my advisor's vision of my project. My advisor seems to have his own vision and ulterior goals that are not completely aligned with mine. I think a good PhD student should just listen to him like a father-figure and do whatever I am told. But I don't agree or sometimes don't even care about his specific vision. For example, why should I do a research on the dorsal fin of a seahorse (for your "dorsal fin grant") when I can study pipe fish instead? I understand that he is the one who is funding me, but I also care deeply about my own academic freedom.


Given these habits, am I doomed to fail my PhD? Do I have an anti-research personality?
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)
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.
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.
This behaviour is fine. One can get a certain fatigue when working on certain projects for a while. 
&amp; 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.