A question about appositive.Do you think this two sentences correct?
[There are two activities in this class, climbing trees and crossing the river.]
[There are two phrases in the book, "move forward" and "go ahead".]
Some people think maybe an appositive cannot leave the modified word, so we can't write like that. We only can write-[There are two activities, climbing trees and crossing the river, in the class]
[There are two phrases, "move forward" and "go ahead", in the book.]
Do you agree it?

Question

A question about appositive.Do you think this two sentences correct?
[There are two activities in this class, climbing trees and crossing the river.]
[There are two phrases in the book, "move forward" and "go ahead".]
Some people think maybe an appositive cannot leave the modified word, so we can't write like that.  We only can write-[There are two activities, climbing trees and crossing the river, in the class]
[There are two phrases,  "move forward" and "go ahead", in the book.]
 Do you agree it?

Answer

Your first two phrases...

There are two activities in this class, climbing trees and crossing the river.
There are two phrases in the book, "move forward" and "go ahead."

...are grammatically correct but require more information at the end in order for your listener to understand your full meaning.

Your second set of phrases are awkward and a bit unnatural.

Answer

The sentences are fine. Instead of "crossing the river", it should be "crossing rivers" unless you have one specific river in mind. If it is one specific river, give its name and capitalize it.

Since you are introducing a list, you can use a colon instead of a comma:
"There are two activities in this class: tree climbing and river crossing."

Answer

They are ok the way they are. Your listeners just need a little more detail. For ex) There are two activities in this class that we are going to do this week: climbing trees and crossing the river.  (By adding the extra little detail, you explain to your listeners why you started talking about trees and rivers in the first place)

Answer

The first looks better because you are keeping the parts of the sentences together.  In this class is a prepositional phrase, so you are ok to put the appositives after this.  It is always good to check for those dangling modifiers. :)