I'm not a teacher, but:
The "Genitive" case of a noun means the possessive case i.e. love's spirit means "the spirit of love". "John's pen" means the pen that belongs to John.
It has no connection with "genital", if that is what you thought.
Other cases of nouns/pronouns are typically:
"Nominative" (He/She) for the subject of a verb.
"Acusative" (him/her) for the object of a verb and with prepositions in English
(plus, in some languages, a separate "Dative" and/or "Ablative" used with prepositions and
"Vocative", used when addressing someone, as in "Oh! Caesar" - nearest equivalent in Arabic would probably be "Saeedi" as in "Dear Sir").
2a) I would tend to use "have improved" for the immediate past and "improved" over a previous period..
I would have no problems with "the weather conditions"
I would only have used "during the last days" in the context of, say, "during the last days of the month", but I would probably have included the word "few" even then. Basically, it needs a qualifier.
2b) Similarly, I would have used "has been/over" for the recent past period and "was/during" for a specific earlier period (say the 80's)
2c) "He should him"?
Time/the time could be interchangeable, but, in using "the" with the children, I would tand to leave out the "the" before time, to avoid repeating the same word too many times in a sentence - it's just my style. I would also only use "from" rather than "at" in this context, because of the "not having time" aspect and "from" implying motion rather than position.
No - maths does not need to be capitalised.
Your comment about American vs British English is correct. The term "math" grates on me!
"both" demands a qualifier such as "both A & B", You wouldn't say "the first was the easier of both A & B", but you would say "... of the two"
I wouldn't have a problem with any of the suggestions, except I would probably leave out the "her", as she wouldn't be leaving on anyone else's holiday etc.
5. This is the other side of 2c), where there is no timing concept involved, so all forms would be OK by me.
Student or Learner