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.