Don't panic!

(1)
As both products are good, order the CHEAPER one from the MOST competent of the two clerks.

The test-taking trick here is to notice the words "both" and "two".
Of the possibilities cheap, cheaper, cheapest, cheaper goes with both /two, cheapest with three or more. Of these: many, more, most, more goes with both/ two.

As both are good, get the cheaper from the more competent of the two.

(2)

There are different ways of handling the apostrophe question. Be sure you understand which standard your teacher is following.

If singular, add apostrophe "s". John's Charles's child's, woman's, secretary's
If plural and it doesn't already end in "s", add apostrophe "s". children's, woman's, men's
If plural and it does end in "s", add apostrophe only: secretaries', managers', workers'

Note that secretarys' is impossible. Since it's spelled as a singular (no ies at end), it has to follow the rule for singular: secretary's.
Likewise secretarie's is impossible. Spelled ies it's plural and follows the plural ending in "s" rule: secretaries'

(3) "Double negatives" is two (or more) negative words in the same sentence.

The negatives are anything ending in n't: don't, wouldn't, doesn't, etc. and the words no, not, no one, nobody, nothing, nowhere, never, none, ..., and hardly, barely, scarcely.

They don't have no bananas today. (don't, no)
We cannot never do a thing like that again. (cannot, never)
We don't hardly have no time to finish none of our tasks. (don't, hardly, no, none)
Don't you have nobody to help you? (don't, nobody)

Hope this helps.

Your example of a double negative is fine.

Sentences 1 and 2 have problems. 3, 4, and 5 are fine.

Concentrate on this: If no possession is shown, you don't want an apostrophe at all!

In the first two sentences I see various words that end in 'S'. But that doesn't mean they all show possession.

Possession is a loose term in grammar.
It may mean true ownership: My house = The house belonging to me. John's house = The house belonging to John.
Or it may mean something less - just an association of some kind: Susan's song = The song that Susan likes; the song we associate with Susan.

So, thinking of "belonging to" in a very, very loose way, in Sentence 1 you have the secretary 'belonging to' the attorney, and in Sentence 2 you have the conclusions 'belonging to the president. So 'attorney' and 'president' are the only words that need to have those "apostrophe 's' " rules applied.

When you put the apostrophe on "witnesses" you made it say: "belonging to the witnesses". I don't think the sentence was saying: The one(s) belonging to the witnesses gave their names ...
That doesn't make sense.

But the second sentence IS trying to say: These conclusions are (the ones) belonging to the president, so "president" needs to be looked at for what "apostrophe 's' " rule applies.

In the last sentence nothing is such that it belongs to the offices, so you certainly don't want any kind of apostrophe treatment there. You may think that these are offices 'belonging to the corporation', however. In that case you would need: the corporation's offices. But, no, in this case the adjective form of corporation (corporate) is used, and that alone is enough to show the "belonging" idea -- so no apostrophe on 'corporate'.

Mr. Jones' progress (American English, frowned upon by some Brits)
Mr Jones's progress (British, and used in the US as well)

Mr Joneses' progress (correct if there are at least two men whose name is Jones)

Cheers
CB