Do all submariners need DV clearance and would having a close relative with a long list of criminal offences (numerous different offences including theft, street robbery, arson and driving convictions) prevent me from being able to join/get clearance?
Probably not. The first rule of DV - admit everything, omit nothing. As long as you follow the first rule, aren't married to someone from Russia or China, then you should be ok. The problems arise when you don't admit something, which can then be used to blackmail you, and/or demonstrate you lack the integrity and moral courage to be entrusted with some of our deepest secrets.
Submariners, same as all Naval personnel are security cleared. DV tends to be for those working in jobs where they are required to gain access to higher security classifications beyond Secret. Communications, being a prime example.
Generally relatives with a string of convictions will only be an issue if the offences were committed in connection with politically motivated nefarious deeds, terrorism or trafficking and the individual has a significant influence in your upbringing which could bring into question your integrity.