بسم الله الرحمن الرحيم
(1) It is makrooh to kill mosquitoes with electric rocket as it results in burning a living creature unnecessarily. It is forbidden in Hadith to burn a living being. It is unnecessary as there are other effective methods to save oneself from mosquitoes i.e. mosquito net, webby / netlike doors, Kachhuwa Chhap or All-Out etc.
(2) Yes, the woman my wear the hair bun (Juda) on the nape of the neck (guddi) it is lawful, however she should not wear it over the head.
(3) Long hair is an adds to the beauty and adornment for women. Hence if there is no difficulty in managing the long hair then it should not be cut down. And if there is severe problem in managing it then some hair may be cut below the hip then it is allowable provided it does not resemble kafir and fashionable women.
(4) Yes, you may use such henna. It is lawful. However, if the excessive use of such chemical henna is harmful and injurious for skin, one should avoid using it very often.
Allah knows Best!
Darul Uloom Deoband