For Physics and Engineering yes, but not so much for Chemistry.
Yes, quite a lot.
Because they are based more on theory than numbers and facts
Lots and lots and lots. Just off the top of my head, anything that uses quantum physics is based off of Hilbert space theory and operator analysis, which are both (older) fields of math.
The equations of motions.
No, mathematics and writing skills are completely separate skill sets, most people are either good at one or the other, few are good at both. It's a good idea to work on your writing skills but in most math based jobs like accounting and engineering, you don't need writing skills as much as your mathematical skills.
Yes, quite a lot.
Physics, Chemistry, etc. Later on in many fields of Engineering and many science-based fields.
Many fields of engineering are based more or less on chemistry: chemical engineering, metallurgy, foods industry, mining, etc.
Generally the opposite is true as chemistry and physics are heavily mathematically based
Physics is a type of Science such as biology and chemistry. Physics is the most math based Science. It can involve anything from gravity to moments (turning force).
Chemisrty isn't based on math, but it certainly wouldn't be possible without math.
Well Physics is the ultimate science but it has to be based on maths to be understood easily. It is not that it can not be understood without maths but mathematics makes it easy. For example if i say that gravitational force between two masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between them and otherwise in maths we can write directly proportional to m1m2 and inversely to r^2. Its not possible to write here with the symbols but the point i m trying to make is that its easy to write and represent it with maths but there is no question physics is the ultimate science and maths is just a tool to make it easy.
Standard physics and chemistry are based on the energy and matter that we know. We do not know much about dark energy or dark matter, but only that it may exist.
Assuming you're talking about university...in the US...Pre-engineering courses are those that ready you for sophamore and junior level engineering courses. For instance, calculus based physics, linear algebra, chemistry, statics, etc. This differ from courses a liberal arts major would take in the amount/level of math they require.
Basic chemistry and physics would be the only specific subjects; otherwise, an electrical technician is mostly taught the regulations, building codes, identification conventions and how to work safely on the job in an apprenticeship or through a trade school. An electrical engineer would need a bachelor's degree and would study subjects such as calculus, differential equations, calculus-based physics and other higher-level science classes.
Activities performed by a chemist that are primarily based on an understanding of physics are any sort of analytical chemistry and instrumentation. In reality, all chemistry is based on an understanding of physics - but specifically physics as it applies to atoms and molecules, work and heat - and thermodynamics in general.
The link between chemistry and Biology is much more obvious than that of physics and chemistry. Every reaction within an organism is a chemical reaction. Every biological being is chemical. That is why a large branch of biology is called biochemistry. In my opinion, there is more of an effect of physics on chemistry than chemistry on physics. What I mean is that physical parameters often dictate chemical reaction kinetics. Also, areas such as electricity (Movement of electrons/positive holes) is heavily chemistry based, more obvious in batteries