I think some of what's been said is a little misleading. A lot of people have implied that because physicists have the ability to rely on experiments, they don't have to worry as much about formal proofs. But this isn't really right, and although ultimately if a particular theory is "correct" or not depends on this, physical intuition can still lead to good ideas and good math, even if it leads to empirically incorrect ideas (in fact, many good theoretical physicists care as little for experiments as they do for proofs!). The history of physics is actually full of "recycling" good ideas from places where they didn't work, to new places where they do! These ideas are so re-usable not because they were based in experiment, but because the kind of reasoning behind them was qualitatively good, even if it did not end up being quantitatively good (compare to similar instances in math!).
I'm a theoretical physicist, but have an undergrad degree in math, and that's caused me to put a lot of thought into exactly how my thinking in my physics training differs from my thinking in my math training. And I really think that the kind of qualitative thinking that I see a lot of mathematicians use to reason with before building a formal argument is almost exactly the kind of thinking physicists use.
I know one of the things made me first realize this, was a few years ago, reading a blog post by Terry Tao. The post was about some analysis topic that I wasn't familiar with (which I no longer recall specifically) and I had stopped to think carefully about what he was saying for a few minutes after seeing a confusing statement, and tried to reason through it using my physics intuition. After getting some idea of what was going on, I finished reading the article, and, after finishing it, I realized that the logic behind the article as a whole (as opposed to behind each individual statement) was basically identical to what my physics explanation was.
In terms of the points you mention above, "physics intuition" would correspond roughly to 1, 5, and a lesser extent 4. But this (from my point of view) seems to correspond pretty well to exactly how mathematicians think both before they formalize an argument carefully, and in the "big picture" point of view (which is really partly inherited from the former).
So in a sense, physical intuition is everything you do in math, up to, but not including the final step when you make your arguments careful. Although we usually go "most" of the way to making an argument careful, ultimately we do have to bring things to the level of being able to make a calculation which one could compare with experiment, and this requires being fairly careful about the reasoning we use being mathematically sensible (although, from the point of view of most theorists, this is not the interesting part).
We also like to break our arguments up into "fundamental" pieces, but not in the same way as mathematicians do, in terms of axioms/definitions/theorems/lemmas, but in terms of "physically reasonable" pieces, since they are easier to get a handle on in terms of theory-building. But the problem is that, while these physically reasonable pieces usually correspond to simple physical statements, they usually correspond to very complicated statements when spelled out axiomatically, which makes that form of them too cumbersome to work with.
It's difficult to explain specifically what the similarities I'm thinking about are, so if you want to see some specific examples, that would be more amicable from a mathematician's point of view, it could be valuable to grab a text on the calculus of variations and go through some of the proofs of things that you already know through other means (e.g., geodesics) since this specifically is one kind of reasoning that's used all over physics. There're also a number of such books written from a solving physics problems point of view.
There is also "quantum fields and strings: a course for mathematicians" which is a bit tougher, but written by actual physicists, and I think could give a good deal of insight into how we actually think. I would avoid anything called "quantum mechanics for mathematicians" for this because they tend to not be written by people who are primarily physicists.
You could also go back to Euler or Gauss or Riemann, since a lot of their arguments are very "physical" and are highly recognizable for physicists. I believe Spivak's volumes on differential geometry contain some of Riemann's papers, along with discussion translating them into modern language which could be useful to see. The MAA also has a "how Euler did it" column that could be interesting in this regard, too.