What are some guiding rules as to the use of articles and demonstrative pronouns (and other linguistic elements like these?) in German?
Honestly, this is a very broad question. If you want to get technical and dive into grammatical theory, the keywords you'll want to look up are determiners and the determiner phrase. Suffice it to say, in German we >>really<< love our determiners, as most of any given noun's grammatical features (case, number, gender) are encoded within the determiner and can only be seen there.
The only case (I can think of right now) where we don't typically use a determiner (and even then, linguists like to analyze this as a null-determiner), is when we are talking about generalisations, uncountable nouns or indefinite amounts:
Probieren geht über Studieren. Ich trinke am liebsten Wasser. Elefanten haben ein gutes Gedächtnis.
So here's your "guiding rule"; if you have noun, you'll want to use a determiner. Now, on to your second question.
Why "das Studium" as opposed to "Studium"?
I think the real root of your confusion here, is that English and German analyze the studying in your example very differently. In English, we have a gerund, which is an infinitive verbal construction and thus can't take a determiner. The German translation however uses a nominal phrase to express that same part of speech. And that means, as we are talking about a specific instance of Studieren, that we need some sort of determiner - in this case, the definite article.
We could actually just rephrase the English sentence to get a very similar nominal phrase that would also require a determiner:
Jazz helps me focus on my studies.
Jazz helps me focus on studies.