Learn English – Students appearing at/in/for the written examination

"The cinema" is correct.

Whenever a noun is used in English in the singular, unless it is a proper name like "Bob" or "Russia", it must have either an article (a/the), a possessive (my/Sally's/etc), or one of a few other special adjectives, such as "one".

You could say, "Jason goes to a cinema every Friday." That would clearly indicate that it is not some specific cinema. But in practice, we usually say "Jason goes to the cinema every Friday." We use the definite article when speaking of something in the abstract, when we are talking about an undetermined member of a class. "Jason goes to the cinema every Friday." "Always look both ways before crossing the street." "The cat is a curious animal." Etc. Logically, by textbook explanations of "a" and "the", it sounds like you should use "a" because we are not talking about one specific instance but any instance. But in practice we use "the".

I think someone made up this rule to make it harder for foreigners to learn English. :-)

as much or even more than is OK, but your example should read:

He feels his troubles as much or even more than they do.

In this way, it is clear that nominative case is correct.