Your sentences 1, 2 and 3 are fine.
I would probably say "I lost it somewhere" or "I left it somewhere" for number one, but "mislaid" is fine - a good word (and probably more true than 'lost').

The wife's original sentences are a bit 'odd' though.

"Many" is often used in negative sentences "There aren't many apples left, can you go to the greengrocer and get some more?" or "I don't have many pens, but you can borrow one."
In positive sentences, "a lot of" or "lots of" would be more common.

The "future tenses" in English have rules upon rules. I don't know them well.
However "going to" doesn't sound quite the right one.
I would expect "Today you're going to a lot of places on business."
(In fact, what I'd expect to hear (as a fairly casual comment) is more like "You're travelling all over today."

"Afternoon" is a noun. "After noon" is a prepositional phrase, which acts as an adverb (at some time later than 12 noon).

You have lots of choices for the second sentence. Your use of "The weather says" to mean "the weather forecast is" is a very common British idiom - it sounds very natural to me. I'm not sure whether it's the same in the USA or elsewhere.
I think that "will stop" is the best tense in most sentences.
The weather says the rain'll stop this afternoon.
The weather says the rain'll stop after noon.
The weather says the rain'll be stopping this afternoon.
The weather says the rain should stop this afternoon.
The weather says the rain should stop some time after noon.

They're all good sentences and would not sound unusual at all. The first is probably the most common.

The use of "should" in the last two gives a slight implication of "That's what the forecast says, but don't be so sure!"

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Wyrd bið ful aræd - bull!