Hello Aspirant ,

Yes you can very well become a law professor after completing distance LLM but for that you will have to appear for and clear NET/ SET exam . To be eligible to write NET exam , you should have a valid masters degree from a UGC recognized university. If your university is well recognized then your distance degree will be treated equal to a regular one .For research fellowship there is an age limit of 30 years and for the post of assistant professor there is none.

Go through this article to know more on the eligibility criteria for NET exam :

https://competition.careers360.com/exams/ugc-net

All the best