Learn English – Next year. Vs in next year

Here are several sentences that a site says is wrong

That site is wrong. These sentences are perfectly fine and used often. Yes, they are not specific date/times, but that doesn't mean they are incorrect.

In the workplace and school settings "next week" usually means "sometime during the next calendar week". So Saturday will come, and then Sunday, and then will be the start of the next calendar week, Monday! And that will end on Friday (usually, unless most people work on Saturday there too).

So common descriptors that would go before the words "next week" would be

• "early next week"
• "later part of next week"
• "middle of next week"
• "sometime next week"
• "by the end of next week"

So when people don't say any more description than "next week", I'd usually take it to mean anytime during next week.

So, I changed it to make it clear sentence like followings.

The next week after this will be sunny and warm.
Give me a call the next week after this one and we'll have lunch.

This is really not making things more clear. It is uselessly redundant. It doesn't make things any more specific than just saying "next week".

(1) seven days starting now

No. Hardly ever would it mean that. If someone wants to say that it will be sunny and warm for the next 7 days, they'll not say "The next week will be sunny and warm." They would say something like, "At least the next 7 days are going to be sunny and warm." Or "The rest of this week and most of next week will be sunny and warm".

So if you want to be exactly clear about what you mean, then just use sentences like this:

• The upcoming calendar week calls for warm and sunny weather.
• Not next week, but the next should finally bring in some good chances for rain.
• A week from today the assignment is due by 3pm!
• Give me a call sometime next week and we'll have lunch.
• She'll be out of town all of next week.
• She'll be out of town till Friday. (or whatever day she'll be back if not gone the whole week)

The number of students appearing for the written examination increases every year.

I would use for to indicate that the purpose of the "students appearing" is to take the written examination.

9. preposition
   You use for when you state or explain the purpose of an object, action, or activity.