Well, my first piece of advice is to ignore everything Microsoft Word says about language. It’s an unreliable source.
“Analyses” is the plural of “analysis”. As with any singular-noun/plural-noun pair, it is possible to construct sentences in which both are grammatical:
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I saw the tree.
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I saw the trees.
Big deal. And note the difference in meaning between the two sentences. What further complicated the matter in the case of “analysis”, however, is that “analysis” can be both a count noun and a mass noun. As a rule of thumb, a mass noun is a noun that can go with “how much”. A count noun is a noun that can be pluralized:
3a) How much stuff do you want? (stuff - mass noun)
3b) How many stuffs do you want? (no good: stuff - not a count noun)
4a) How much furniture was there? (furniture - mass noun)
4b) How many furnitures were there? (no good: furniture - not a count noun)
5a) How much dog was at the park? (no good: dog - not a mass noun)
5b) How many dogs were at the park (dog - count noun)
The interesting thing about some nouns is that they can function both as count nouns and as mass nouns. “Analysis” is an example of this:
6a) How much analysis of the data did you do? (analysis - mass noun)
6b) How many analyses of the data did you do? (analysis - count noun)
So in your friend’s sentence, “viewers of the movie would have to go through analyses to find any emotional…”, “analyses” is obviously being used as a count noun, since it’s plural. Your correction to “…have to go through analysis to find…” would give it a mass-noun interpretation. Because “analysis” can be both a count noun and a mass noun, there is not too big a difference in meaning between these two formulations.
In regard to your two example sentences:
It was suggested:
Not necessarily. In the first case, maybe John and Sarah both did a bunch of analyses together. But I agree that in the second case, John and Sarah necessarily worked together.