First, let us review the adjectives involved to make sure there is no confusion.
Size vs. Quantity/Frequency:
Size:
「[小]{ちい}さい」 = "small".
「[大]{おお}きい」 = "large"
Quantity/Frequency:
「[少]{すく}ない」 = "few", "a little", "not frequent"
「[多]{おお}い」 = "many", "much", "frequent"
Moving on to Grammar:
To express "to get or become (adjective)" in Japanese, you need to attach the verb 「なる」 to an adjective. In doing so, however, you need to conjugate the adjective into its [連用形]{れんようけい}(conjunctive form). Yes, just like verbs do, adjectives also conjugate in Japanese.
The 連用形 of 「小さい」 is 「小さく」. Likewise, 「少ない」 to 「少なく」.
「小さくなる」 = "to get small(er)", "to shrink in size", etc.
「少なくなる」 = "to lessen in quantity", "to become less frequent", etc.
連用形 is extremely important. If I, who is neither a teacher nor linguist, must use the word on a daily basis, that should tell you how important it is.
The plain negative form of an i-adjective is also formed with 連用形. 「小さい」 becomes「小さくない」 and 「少ない」 becomes 「少なくない」. All humans live in the world of 連用形. In Japan, even animals live in the world of 連用形.
Finally, your sentences:
「[飲]{の}んでいるでしょう。その[証拠]{しょうこ}に、ぜんぜんお[腹]{なか}が[小]{ちい}さくならないじゃない。」
"You must have been drinking (alcohol). As evidence, your belly hasn't shrunk one bit!"
「その証拠に、[実際]{じっさい}に[作文]{さくぶん}の[宿題]{しゅくだい}をチェックしたら、[正]{ただ}しく[書]{か}けていないことが[少]{すく}なくない。」
I skipped the first sentence because your translation of that is good.
"As evidence, when I actually checked my composition homework, (I found that) there are quite a few things that have not been written correctly."
「正しく書けていない」 literally means "has/have not been written correctly". For a translation purpose, one could use "I have not written (or even 'did not write') correctly."
「少なくない」, which expresses high quantity/frequency, is one of those phrases that we native speakers use far more often than Japanese-learners do. It means 「多い」, which is what the learners tend to use.
