The primary goal of problem management is to reduce adverse impacts caused by incidents and to avoid the recurrence of problems related to these incidents.
Problem Management is engaged when,
Yes, this is possible. Choose the option 'Close all associated incidents' from Setup > Automation > Closure rules > Problem Closure Rules > Optional Rules.
You can automate in such a way that, when a solution is added to the problem it automatically gets copied to every incident which is associated to the problem.
This can be done from Setup > Closure rules > Problem Closure Rules > Optional Rules, and selecting the option 'Copy problem solution and workaround to all associated incidents'.