The clue is to hear if the s is voiced or not. Ils ont - voiced s (z), but ils sont - unvoiced s (s). The s is pronounced voiced when it comes between two vowels.
In any case, you can usually tell by the grammar, like with this sentence. Since ils is masculine and chemises is feminine, the sentence would be ungrammatical with sont
no, he = il, they (for a mixed or masculine group) = ils. Verbs have different conjugations for these, so even though you can't always hear the 's' in 'ils' (they) you know by what verb is said with it. so here, it is ils ont = they have. if it were 'he has' then it would read 'il a'.
In french, ALL nouns must have an article. In English, this isn't the case, as with plurals it's grammatically correct to choose whether to have the article with it. In french, it isn't grammatical to say just 'chemises', you should say 'des chemises'. Even though it's literally translated to 'some shirts', when put in context, it's used in the same way English would just use 'shirts'.
How to figure out whether "des" means "some" or is simple the short form "de+les" and may be translated with the article "the"? In some tasks its totally fine to use that article and then again its being marked as wrong, like in this task.