You're right about the first point, Anna.

1.  There is a mistake in the article: 

Go over something: to feel better after an unpleasant event or an illness

This is wrong. As you pointed out, this should read get over something, not 'go'.

There are often inaccuracies in the articles published here. Italki makes sure that articles look nice, with pretty photographs and so on, but they don't always check or edit them properly.

2. The original sentence is fine. Your suggestion sounds awkward and confusing. You can't have such a long chunk of language between the two parts of the phrasal verb.

‘To go over’ something is more like to revise, and ‘to get over something/someone’ is to move on from something. 

‘Let’s go over the main points of the lecture again’

’she got over him quickly after they broke up’

`Example: It took me a while to get over the break up. Even now, I still think of him sometimes.`

Dictionaries tell me that `to go over` has a different meaning. So, now I'm confused.

2. `I tried using the phrasebook, but I just couldn’t get the idea across that I didn’t eat any meat.` sounds a bit weird to me. I would rather say: `I tried using the phrasebook, but I just couldn’t get the idea that I didn’t eat any meat across.` It makes more sense to me. Is it possible to say it this way?

Su.ki. Was correct of course, but in American English, people will give you a lot more slack on sentence 2.

I couldn’t get the idea across“ is far more common,  but „I couldn`t get the idea that I didn’t eat meat across“ works for me too and doesn’t sound odd or in any way awkward. It could be that because I speak German as will as English that kind of word order sounds natural to me in German as separable verbs wind up with the separated portion being placed at the end of a sentence anyway (in fact has to).