The preposition без (takes genitive case) means "without." The root is плат- (плач-): Я не плачу. Он платит. (I am not paying. He pays.) In a prefix terminal з (a voiced consonant) becomes the unvoiced с before an unvoiced consonant (for example, p, t, f, k in contrast to b, d, v, g). The suffix -н- is an adjective marker. The short-form neuter singular adjective (-o) is very often used as an adverb. Several prefixes can be used with the noun form плата.
Он работает без зарплаты! He works without wages! Он переделает работу без доплаты. He will redo the work without additional payment. Это бесплатное окно. This is a free window. Это бесплатно? Is it free? Да, вы получаете окно безплатно. Yes, you are getting the window for free.
The TANSTAAFL principle is simply expressed in Russian: Нет бесплатного сыра!
It's either accusative or instrumental cases. A couple of links: http://masterrussian.com/vocabulary/za_preposition.htm https://en.m.wiktionary.org/wiki/за-
You can, but there is a subtle nuance: "He writes articles for a living" means that it's what he does, earns money by writing articles and lives off it. "He writes articles for money" can mean the same, but can also have negative connotation: he takes money to write ordered articles (or puff pieces).
Note that для and за have very different meanings. Also, "for" has very different meanings. For example, "for" can mean "in order to give to" and can mean "in order to get" (give and get are opposite in meaning).
Он пишет статьи для газеты. He writes articles for the newspaper.
Он пишет статьи за деньги. He writes articles for money.
Он пишет статьи для газеты за деньги. He writes articles for the newspaper for money.
From what I've gathered:
за is used when an action is being done or an object is being used to attain a result or goal (Он пишет статьи за деньги = He writes articles 'for the goal of' money)
для is used when an action or object is supposed to affect or be used by someone/something else, but it's not necessarily for a higher purpose or goal (Я купил бензин для машины = I bought gas for the car)
За has a handful of other uses and definitions too. Like many Russian prepositions it doesn't seem to have a single English equivalent.
In Katzner's Russian-English dictionary, за has a huge number of entries: In accusative case, there are 15 different, somewhat related entries, and in instrumental case, there are 10 different entries. It literally is one of the "most defined" words in this dictionary, because it has so many different usages.
If you're serious about learning Russian, this dictionary can be quite helpful.
In this case the translation is into -indefinite plural-, and English has a 'zero article' for that; in the indefinite singular it would have "a(n)".
I'm pretty sure the Russian could be equally translated with the definite instead: "He writes the articles for money," IF we assume there was prior discussion of these articles.
There are rules that cover the bulk of cases, but as always there are tons of exceptions:
"the" is used in a definite noun phrase (singular or plural) that doesn't already have another 'definite determiner' (such as "my, this, __'s")... except not for most proper nouns
"a(n)" is used in indefinite noun phrases when they are both 'singular' and 'countable'. "a(n)" derives from an unstressed version of "one" and can still be used much the same way (except that it is obligatory)
the 'zero article', which is simply nothing, is used for indefinite noun phrases that are plural and/or uncountable (you can optionally use "some" in these cases), as well as with most definite proper nouns
This article gets into more details and has examples: https://en.wikipedia.org/wiki/English_articles
I don't know whether you used Google to translate English to Russian or Russian to English. I often use Google to save some time, but I almost always find bad errors in the Google translations. In most cases, it is faster for me to correct the errors in the Google translation than to type out my own complete translation from scratch.