These are indeed joint questions and to be answered interleavingly.
As I have commented, it is not really that "at the end, important (new) information should be placed". Rather, in all languages, there exist means to contrast different kinds of information expressed in a sentence in the context of a discourse. The linguistic discipline that deals with these phenomena is called information structure or information packaging.
Now, depending on what you want to say, there are different possibilities what "important/new" might mean, and how that can be put into speech1:
Den Franz, den hab ich gestern zersägt.
Den Franz hab gestern ich zersägt.
Gestern hab ich den Franz zersägt.
Zersägt hab ich den Franz, gestern.
In German, as in English, the main means for information packaging are syntactic and certain phrasal constructions (typical means are dislocation, clefts, stress in spoken language, or phrases like "as for X, ..."). You can see that not always, "emphasized" things are put at the end; although in general, left and right dislocations are often employed for information packaging. Therefore, the answer to question one is "sometimes, but not really, and neither in English".
If that sounds interesing, I can very much recommend Martin Hilpert's lectures (they're in English).
Now, for the terminology: there are traditionally three "axes" in which packaged information is divided:
- Topic & Comment (Topik & Kommentar): this is the distinction about what is talked about currently, and the information given about those entities. Think of a story or play, where you have some characters and their actions. Classical fairy tale: "There are three little pigs, they live somewhere. In their little house, they do . One day, the wolf came to the pigs." (Avoiding topicalization is one of the reasons why languages have a passive voice.)
- Focus & Background (Fokus & Hintergrund): while discourse is essentially a sequence of propositions, for each proposition there is usually a certain part that is the actual thing you want to convey, and the "rest around it". Look at the last two example sentences: they express the same situation, but in the first, the patient is hightlighted, whereas in the second, I wanted to focus on the action.
- Given & New (untranslated): during discourse, you can inroduce new entities that are referred to. This is the reason pronouns exist: you introduce things by name, and can then refer to them shortly as already known.
You see that these naturally correlate: a new entity is usually focussed. Topical entities are by default "somewhat focussed", unless you explicitely focus elsewhere. Background and comment usually overlap. But all axes are in principle independent, and that is why there exists so many variations in contrasting parts of information.
As a short excours: some languages even have morphological means to mark information structure -- often by "topic markers". I think Tagalog and Bikol on the Philippines do, and there is this famous Japanese example:
Boku-wa unagi da
As for me, it is eel.
which you can say when you order at a restaurant -- and not to call yourself a fish. Boku is the pronoun "I", -wa is the topic marker, unagi means eel, and da is a copula.
1Don't worry, this is just my standard maximally transitive example sentence. And the Austrian habit of using articles with names is practical to emphasize case distinctions, and makes things sound more natural to me.