Insofar as oil & gas - my understanding from friends in the business is that there's room for analytical/physical chemists for analysis and characterization of materials from extraction to final products, as well as chemists of a more synthetic bent, focusing on catalytic methods, polymer chemistry, process chemistry, and various other odds and ends. From my perspective, the oil & gas industry is basically one big playground for one to apply organic chemistry, but at a rather large scale.
I suppose it matters on how one defines "sparse" - undergrad chem majors in the US, in my experience & observations, usually run through the same typical lower division math sequence that physics and engineering majors are expected to do (single & multivariable calculus, introductory linear algebra, and an intro-level differential equations course), although aren't typically required to do any more. It's usually more than what biology majors are expected to do, although clearly it's nowhere near the amount that physics and engineering majors do. In principle, one can graduate with a chemistry degree and consider everything from biochemistry to physical chemistry to synthetic organometallic chemistry a legitimate option. In my observations, the people who were more inclined toward theoretical chemistry usually had a double major in physics or mathematics to go along with their chem degree.
Insofar as coursework for physical/theoretical chemistry research - it depends. My anecdotal experience is that chemistry departments don't have the same inclination for making students crank through as much coursework as physics departments, but YMMV on that. There's usually some coursework outside of the field (whether official or via audit), but there aren't any standard rules for it. I took a year of formal classes and another year's worth of audits during my time in grad school. That applies to most of my friends in the chemistry program had the same sort of schedule. Most of my acquaintances & friends over in physics were in formal coursework for at least 1.5 to 2 years. Of course, if one is in an interdisciplinary chemical physics program, the course requirements are defined and are generally slightly more extensive than a typical chemistry graduate program.
The other thing to keep in mind is that there can be quite a bit of self-learning (obviously!) and "internal" classes. My doctoral work involved solid state NMR, so my research group had semi-regular tutorial sessions to get the new people caught up on the essentials as well as explore the more recent developments of note. It's not a topic which is conducive for a department to set up a formal class (as it's really only of significant interest to those of us in that research group).