I urgently need you to check a few more sentences (infinitive/gerund) . Thank you very much.

1) After that, we can establish an e-twinning-platform so that our students can start emailing to one another. (is "establish" possible?)
2)I can’t afford to buy new shoes.She arranged for her mother to be driven to the airport.
3) She chose not to buy the shoes.They didn’t dare to walk into the camp.
4) She is planning to go to the seaside.
She admitted cheating during her maths test
5) She advised/suggested/recommended calling for the doctor.
6) She always avoids being tested orally.
7) They discussed selling their house.
I don’t feel like eating an ice-cream.
8) She imagined winning at the lottery.

1. Yes, "establish" works ... or you could use "set up" if you wish.

2. OK

3. ... didn't dare walk into the camp.

4. OK

5. delete "for"

6 - 7 = OK

8. delete "at"

1) After that, we can establish an e-twinning-platform so that our students can start emailing to one another. (is "establish" possible?)

Yes, "establish" is possible in this context. In this sentence, "establish" is used as an infinitive verb to express the action of creating an e-twinning-platform.

2) I can’t afford to buy new shoes. She arranged for her mother to be driven to the airport.
In this sentence, "buy" is an infinitive verb, and "be driven" is a passive infinitive verb phrase. The structure "arranged for [someone/something] to [verb]" is a common way to express arranging an action to be done by someone else.

3) She chose not to buy the shoes. They didn’t dare to walk into the camp.
In these sentences, "buy" and "walk" are infinitive verbs. The construction "chose not to [verb]" and "didn't dare to [verb]" are both examples of how to use an infinitive after certain verbs.

4) She is planning to go to the seaside. She admitted cheating during her math test.
In these sentences, "go" and "cheating" are both gerund verbs. The gerund ("-ing" form of the verb) is used after the verb "planning" and "admitted" to express an action that is the object of the verb.

5) She advised/suggested/recommended calling for the doctor.
In this sentence, "calling" is a gerund verb. The verbs "advised," "suggested," and "recommended" are followed by a gerund to express the action of suggesting or giving advice.

6) She always avoids being tested orally.
In this sentence, "being tested" is a passive gerund verb phrase. The verb "avoids" is followed by the gerund to convey the action of avoiding being tested in an oral manner.

7) They discussed selling their house. I don’t feel like eating an ice-cream.
In these sentences, "selling" and "eating" are gerund verbs. The verb "discussed" is followed by the gerund to express the action of discussing, and "feel like" is followed by the gerund to express a desire or lack of desire to do something.

8) She imagined winning at the lottery.
In this sentence, "winning" is a gerund verb. The verb "imagined" is followed by the gerund to express the action of imagining winning the lottery.