grammatical errors in the sentence

You are correct except in the first sentence that is 'Rain is falling'. It will be corrected by only 'It is raining' and it is not present tense but present continuous tense. 'Rain is falling' can never be corrected by ' it rained' due to difference in tense.

Your corrections are, indeed, correct "Rain is falling" is perfectly good English, though not something one might say often. It's more the sort of expression you might find in a poem or a novel. As a general observation, we would usually say "It is raining!". On the other hand, you might say, "Rain is falling on the hills over there but here it's bright sunshine!" Words like 'abroad', 'away', 'afar', 'behind', 'before' and so on have their own prepositions "built in". The 'a-' and 'be-' are remnants of Old English prepositions (meaning roughly 'to' or 'at') which, because the words are so common, have become incorporated into the words themselves, so one never says 'to abroad', 'at afar', 'to behind', for example. However, one DOES use other directional prepositions with them: "He is arriving from abroad", "She called from afar". When talking about time (year, month, week, day, hour and so on). "I met him last week" is correct. If you want to express a certain period of time during which something happened then "I met him in the last week" is correct. In talking about an exam or a test, we normally use the active mood of the verb 'to pass', therefore "I passed [my driving test]" is normally the correct form. If one is speaking of something done to one (specifically by an examiner), one might say: "The driving examiner passed me" (active form) or "I was passed by the examiner" (passive form). Likewise, "I passed the old man on the road" and "I was passed by a black Mercedes travelling at high speed" (passive) or "The black Mercedes passed me just before it crashed into the barrier" (active).

I want to know that what is the age of your students? Yes you have written correctly but one thing is He is going abroad - He is going to abroad. Had is Past and Have is Present.