For this answer, I referenced this article.
Lay vs. Lie
Okay, so this question is dealing with a couple verbs, Lay and Lie. The various forms of the verb in different tenses overlap, so let's just lay them out up front.
To Lay – to put or place something
To Lie – to rest or recline
So let's go through each of the sentences and figure them out.
1. According to our team captain, an extraordinary challenge laid before us.
This one's incorrect, but not for the reason you think. The sentence isn't future tense. It looks like it's meant to be past tense, in fact (when the captain was talking to you, this is what he said). More important, though, we're talking about this challenge resting. No one's putting it anywhere. So we're using "to lie". "To lie" has no form that is spelled like "laid".
So, it's incorrect because it's using the past tense form of "to lay" instead of the past tense form of "to lie". It should read,
"According to our team captain, an extraordinary challenge lay before us."
(This is the most confusing part of lay/lie; "lay" is past tense of "to lie".)
2. Licking her paws, the lioness lay on the shaded grass beneath the trees.
This one is actually correct. It's written in past tense, but using the present participle "licking" to indicate what she was doing when she did the lying.
3. Because he had a bad case of the flu, Pranh laid down all afternoon.
This one is actually just like the first sentence. It's incorrect because "laid" is a form of "to lay", which always needs an object. The correct sentence would be
Because he had a bad case of the flu, Pranh lay down all afternoon.
4. Janine's purse and sweater are laying on the table in the foyer.
Like 1. and 3., this one's incorrect because it's using "to lay" instead of "to lie". The present participle of "to lie" is "lying". So the sentence should look like
Janine's purse and sweater are lying on the table in the foyer.
My Rule of Thumb:
Lay and Lie are very confusing. This is the key difference between the two:
Lay requires an object. You lay/laid/have laid/are laying something (down)—often somewhere.
Lie takes no object. You just lie/lay/have lain/are lying (down).