The specific dictionary entries show that "associate" completely encompasses "socialize", so you can use them interchangeably without fear of being misunderstood:
socialize — take part in social activities: to take part in social activities, or behave in a friendly way to others
- connect things in mind: to connect one thing with another in the mind
- pass time with somebody: to spend time together with somebody
- mix socially or professionally: to be involved with somebody or something in a personal or professional capacity
But "associate" and "socialize" do carry slightly different connotations due to associate's first definition. When describing someone who associates with a particular group of people the connotation is that they are "associated with" that group:
I want to be associated with intelligent people. / I want to associate with intelligent people.
I want to be with intelligent people. / I want to socialize with intelligent people.
This difference is subtle, but here is how it effects two of your specific examples:
I am a positive person and I like to associate with other positive persons.
This person wants to be around and be associated with positive people.
I like to socialize with new people.
This person likes to be with new people. You could use "associate" here and it would have the exact same meaning -- it is unlikely you could really be associated with new people.