Why is the electric field inside a conductor always zero?
By definition, a conductor is a material where charges inside may move freely in response to the presence of an electric field. Suppose such a field exists inside the conductor, then those charges will move as long as there is a field. The redistributed charges create their own electric field, which will cancel the effect of the original field, until the net electric field is zero, at which point the charges will no longer move. So in the presence of an external electric field, the charges inside a conductor will redistribute themselves to create an exact opposite field, giving a net field of zero inside.
How do you know what methods to use when doing an integration?
Integration is challenging. Unlike differentiation, which has definite rules and formulas where you can always get an answer if you follow them, such rules don't exist for integration. You have different methods for different situations, but there are many functions where you need to combine several methods or the method to use isn't clear. To succeed, you need to first be familiar with the basic methods, such as u substitution, integration by parts, trig substitution, etc. You need to practice enough with each method such that you are familiar with the form of function that requires it, and learn to recognize patterns. Beyond that, often it comes down to trial and error, where you have to try several methods and see if one works. Also, you often need to "play around" with the expression, use algebraic or trigonometric tricks to change its form before a way forward is clear. Practice and familiarity is key, and don't be afraid to "get your hands dirty".
Why is dimensional analysis important in physics?
Physics equations give relations between actual physical quantities. As such, two quantities can only be equal if they have the same dimension (units). A good way to check if an equation is valid is to see if units on both sides are the same. When deriving or using an equation, always make sure the units are correct, or you may end up with nonsense such as a kilogram is equal to a meter. Dimensional analysis can also allow you to make educated guess on form of an equation, accurate up to constant factors.