It’s crystal clear that Martyrs are alive and get near Allah their noble sustenance but Quranic definitions and traditions are different from our definitions and culture. We are still very far from Quranic teachings. What definition we do have for the death is not the same as Quran says. We think those who talk, those who eat and drink and those who walk are alive but the Quran sometimes calls such people as dead while those who are not near us and we can’t see them, Quran says they are alive.It means death and life is different in Quranic terminology from our own.
Quran says:
يَا أَيُّهَا الَّذِينَ آمَنُوا اسْتَجِيبُوا لِلَّهِ وَلِلرَّسُولِ إِذَا دَعَاكُمْ لِمَا يُحْيِيكُمْ وَاعْلَمُوا أَنَّ اللَّهَ يَحُولُ بَيْنَ الْمَرْءِ وَقَلْبِهِ وَأَنَّهُ إِلَيْهِ تُحْشَرُونَ

O you who believe! answer (the call of) Allah and His Apostle when he calls you to that which gives you life; and know that Allah intervenes between man and his heart, and that to Him you shall be gathered.(8:24)
As for grieving and crying on Imam Husain (a.s) we have many authentic Hadith in which it has been mentioned that Holy Prophet cried for Imam Husain many years before his martyrdom.
Also we don’t cry that why Imam died. Of course death is reality of life. We all die even Holy Prophet (pbuh) has died but we cry on oppression and non recognition of his place by the Muslims of his time. They committed a very shameful act with him and with his children and women.
Also there are many more reasons for the crying on Aba Abdillah (a.s). One of the very important reason is by crying we do declare that we are not with the tyrants and we refrain from their brutal acts. Also our hearts becomes soft with crying. That is a necessity of a believer that his heart must be soft. A believer can not be a stone hearted person.