He saw the police roadblock but ran through it because he was smuggling drugs. Might be better because at that point in time he had drugs on him and was actively smuggling. Your first sentence 'because he smuggled drugs' makes the sentence more general. So, it could also mean he might not have had drugs on him when he ran through the roadblock, but he might also have had them on him. So the meaning and timing becomes a little vague :).

Run through is the correct verb here. However roadblock is singular,so he ran through IT. I would agree with Alena that using 'he was smuggling drugs' is better than 'he smuggled drugs' in this context. So the full sentence 'He saw a police roadblock, but he ran through it because he was smuggling drugs.'

No

The first sentence the correct one. The reason why is because the word "saw" is in past tense, so that means all the verbs in the sentence have be past tense too. I hope this helped.

Your first sentence is correct. "He saw a police roadblock but ran through them because he smuggled drugs."