Part 2; please read part 1 first.

1/ You could also say "That kind of journalist would...", but you need to stay consistent, choosing either a singular expression or a plural one. I've used plural here because it allows you to say "their reviews". 2/ Generally "praise" is not countable; you can't have one praise or two praises, it's just "praise". 3/ It's not the power that could be bothered, but the people who hold it.

In the 18th century, periodicals in print were either gazettes or newspapers. They went out with a variable frequency which could be each week, every fifteen days or every month and they distinguished themselves according to the topic they dealt with.

"variable frequency each week" would mean every one, two, three or up to 6 days. "which could be" separates "each week" from "variable frequency", making it just one of the options.

This was well written.

News in eighteenth century

1/ You inform someone of something, not with something. "With" might be used with a word like "provided"; for example, "he provided me with the latest news", but not with "informed".

2/ You write in a publication or for a publication. If you write "on" a gazette, you're just scribbling notes on a copy of one.

3/ It's not completely wrong but I probably wouldn't use "the" there. You aren't talking about a specific collection of gazettes there, but rather "whatever gazettes happened to be being published at the time". "...write in gazettes" is therefore enough.

Sometimes the disdainful term of "fogliettanti" was applied to those who in exchange for money wrote in the "fogli" (sheets of paper) what the nobles, the clergy, and the wealthy bourgeois wanted to hear.

1/ The "of" isn't completely wrong but it's not needed here; I wouldn't use it. 2/ Similarly the use of "was" isn't essential, but it makes the distinction between the name simply applying to those people, and it being applied to those people by others. It's not something that "just happened", it was something that other people decided to do. 3/ The last item in a list should always have "and" before it.

For profit, even the "gazzettanti" would even be ready to write the false news.

1/ "For profit" would not have a hyphen. 2/ I think you mean that not only the fogliettanti would write false news, but the gazzettanti would as well. In that case you would have to say "even the gazzettanti". 3/ "False" is an adjective, not a noun, so you can't say "the false". The noun would be "news", or perhaps "articles" or "stories".

These last two terms came into common use, along with "menanti", that is, in Italian, those who sold the pages to the best highest bidders.

1/ I don't think I'd use "the pages" here. Pages are literally the physical objects, which are of course sold to readers. However it could be used as a metaphor for the editorial content of the publication. It's a longer expression, but I'd probably say this as "sold control of their editorial content to..." or perhaps "sold the right to favourable stories..." 2/ The "best" bidder would involve making some sort of value judgement about the bidders. In this case you are talking about the bidder who offers the most money. In English that is said as "the highest bidder". Continued in part 2.