Which one is correct.?

1.The children love to play the forest. ¶
2.The children love playing the forest.

Both sentences need a preposition, eg:
1. The children love to play in/ at/ around/ away from the forest.
2. The children love playing in/ at/ around/ away from the forest.
You could think of other suitable prepositions to use here. These are called prepositions of place (locality).

There is not much functional difference between "love to play" and "love playing". Both the "to play" and the "playing" are acting as infinitive verbs. Both imply that the children usually, and on an ongoing basis, receive pleasure from their activities in relation to the forest. If you wanted to emphasise that "at this very moment" the children are enjoying themselves in the forest, you could say: "The children are loving playing in the forest (just now)." The verb phrase, "are loving" is the present continuous tense, and it implies that the action of playing is current at the time you are speaking.

3. Charlie run to get Dad.

If you wanted to provide a plain statement of fact here, you would say: " Charlie runs/ is running to get Dad." or even "Charlie runs/ is running in order to get Dad." Grammatically, this is called the indicative mood/ mode, as it indicates, states or shows a factual situation.

However, if you wanted to provide a command to Charlie, you would say: "Charlie, run to get Dad!" In grammar terms, this is the imperative mood/ mode, as it expresses your desire/ wish/ need for Charlie to run and get Dad.

4. Charlie run to Dad.

Similar explanations, as above.

5. Zoe and Paul and their parents go to another town to see their grandparents *in the holidays.

I'm assuming that the grandparents being visited are those of Zoe and Paul, but not of Zoe and Paul's parents. Therefore, if you do mean that in the holidays, Zoe and Paul see their grandparents, whilst the parents of Zoe and Paul see their own grandparents too, then your sentence is correct.

However, if you mean that Zoe and Paul see their grandparents, and the parents of Zoe and Paul are seeing these same two people (that is their own parents), then your sentence needs a bit of a change. In that case, you could say: "Zoe and Paul go to another town in the holidays to see their grandparents, and their own parents accompany them on that visit."

I would say that there is perhaps not much functional difference between "in the holidays" and "on the holidays". You are basically saying "during the holidays". Like in most other languages, what is the most appropriate preposition to use in certain circumstances is not always fully clear!

6. Zoe and Paul and their parents go to another town to see their grandparents *on the holidays

See above.

7. We all can go home.

What is being said here is: "We, all of us, can go home." The "all/ all of us" is being used as emphasis.
Your sentence is correct.

8. We can all go home.

This sentence is also commonly spoken as such. There is not much practical difference between this sentence here and the one above. There is a subtle difference, however, because I would say that what you are saying here is: "We can, all of us can, go home."

1.The children love to play in the forest. ¶
2.The children love playing in the forest.

Both sentences are missing a word. Both are correct they just mean slightly different things. 1 means they love to play in the forest generally, whereas 2 implies they are currently playing in the forest and love it.

3. Charlie runs to get Dad.

This means Charlie wants to take his dad somewhere. "go get dad" means "go find dad and bring him here"

4. Charlie runs to Dad.

This means Charlie runs to where his dad is. Neither sentence is wrong they just mean slightly different things.

5. Zoe and, Paul and their parents go to another town to see their grandparents *in the holidays.

This means they go during a period of break, eg in the summer holidays

6. Zoe and, Paul and their parents go to another town to see their grandparents *on the holidays

This means they go on a specific holiday, eg on Christmas, on Easter, on new years day.

7. We all can go home.

I would say this is wrong but it could still be said if you were wanting to emphasize that absolutely everyone could go home.