LLB

Mohit, there is not such thing as degree through correspondence but regular kind. It would be specified on your degree or the name of the institute will state that it is a correspondence degree.
Better would be go for regular one. Entrance tests like CLAT/ LSAT etc are over. So, try for admission in institutes where admission is possible on basis of both I.e. CLAT/ LSAT as well as without it.
Law schools that offer admission through CLAT/ LSAT as well as previous qualifying exam are
Lovely Professional University, Phagwara
Amity University, Noida
G D Goenka University, Gurgaon.
Check their websites for more details.
Good luck.

Yes you can by getting rank in CAT examination. and MBA always is a good option.

No university offer 5 years integrated BA LLB course by distance learning. Also, LLB obtained by distance learning us approved by Bar Council of India and such graduates can not practice in courts.

I am not very sure of this. but I don't think an advocate degree here will work in other country as the entire laws are difficult. You can ask your fellow or seniors for better advice.

B.LLB you can pursue.