1) "Jedu do Česka." and "Jedu na Slovensko" are the only correct versions. We also say "v Česku a na Slovensku". There is no logical explanation for this other than we use the prepositions based on historical conventions, how they were used in the past. You can find some similarities to help you choose the correct preposition, but they never work 100%, so it's better to learn it by heart.
2) "Pomoci" is used in written language, is formal and in spoken language slightly old-fashioned. Until recently it was the only accepted form of the verb. Nowadays, also the newer form "pomoct" is correct and it's preferred in spoken language.
3) The difference is in the verbal aspect. "Hrát" is an imperfective and "zahrát" is a perfective verb.
6) "Opakovat" is imperfective, "zopakovat" is perfective.
8 "Pít" and "jíst" are imperfective, "napít se" and "najíst se" are perfective (and reflexive)
4) It's not totally necessary to use "si" in this sentence, but "smět" is one of the verbs that can be reflexive and it sounds a bit more polite if you use it. You also emphasise the fact that it's you putting the books on the table.
5) It's grammatically correct to use "si" twice as both nesmět and vzít are reflexive, but it sounds a bit odd and we mostly prefer to use only one reflexive pronoun in such cases.
7) Mohl bys mi pomoct se to naučit. or Mohl bys mi pomoct a naučit mě to?
9) Imperfective verbs have all 3 forms (past, present and future; eg. pil jsem, piju, budu pít in nom. sg.), while perfective verbs have only 2 forms (past and present; in this aspect, however, the present is indicating future, because the verb itself is used for one-time, finished actions). Then "jím" (imperf.) means "I eat right now", "budu jíst" (imperf.) means "I will eat in the future", and "najím se" (perfective) means that you'll eat and finish the eating. If it makes sense. The sentence "Alex si je vezme a čte si je." is ok. "Čte" is imperf. so there's no indication if he finishes reading it.