Welcome to Gaia! ::

(It seems to me that we have the same problem since my teacher makes simple problems sound complicated. I really liked Physics in high school, but now err... my teacher is now my problem or more like we don't understand his question. In our long test, we all computed for the angle, but it turns out that his question was the height. It's just that he doesn't seem to know how to properly choose his words in terms that we can fully understand.)

It would be better is you give an example of difficult arc length problems so that we can start from there because your question is too general.

In using radians/degrees, you just have to take note of the given values if it is either in radians/degrees. Besically, the degrees will have a degrees sign and the radians do not have any symbol.

In studying speed it is important to note that if the motion is constant (not accelerating), the equation is displacement over time (also known as the distance travelled over a certain time).
Linear speed is also called tangential speed in our class (imagine a line tangent to the circle because that is how this motion works except that it becomes circular when we see it). Considering that it is not accelerating, the formula for speed would be distance over time. Thus, it is arc length over time.
Angular speed is the speed at which the angle changes. Thus, it is theta over time.

As for your other questions, it would be more helpful if you give a more specific question than a general one so that we may know what specific aspects to talk about.