How does the number of clauses affect the difficulty of a 3-SAT problem?
In general, there is no connection. An instance with a "small" number (say a few thousands) of clauses can be very difficult to solve in practice, while an instance with a "large" number (say several millions) of clauses is easy. It's the structure that matters, not the number of clauses.

For random 3-SAT, you can have a look at the satisfiability threshold.