1. Smoking encourages cancer.
It's grammatical, but it sounds strange to me to use 'encourages' in this context.
2. He was encouraged by the thought of being with his wife again to hang on
in a boat adrift in the ocean.
It's a grammatical sentence, but it would sound better with the ablative clause 'by the thought of being with his wife again' at the end.
3. Banks encourage people to use credit cards.
4. The win over Portsmouth encouraged Bolton in the relegation battle.
5. Refraining from smoking encourages health in people.
Same as 1.
6. Visual aids encourage people to pay attention.
7. Usually dogs are kept to discourage burglars.
8. His parents has always discouraged him from turning pro.
'have always' or 'had always', depending on context.
9. With Agassi and Sampras having retired, tennis fans are discouraged from attending tournaments or Agassi and Sampras having retired has(have) discouraged fans from attending tournaments.
Both sentences are OK, if the second alternative is 'has discouraged'. The noun clause is third-person singular whether or not its own subject is singular or plural - 'Agassi and Sampras having retired' would be rendered by the pronoun 'it' in both sentences.
10. The police firing rubber bullets, tear gas and water cannons to disperse protesters seemed the government's discouragement of subsequent anti-government protests.
This is ungrammatical. A correct alternative would be: "The police firing rubber bullets, tear gas and water cannons to disperse protesters seemed to be the government's way of discouraging subsequent anti-government protests."