What are the implications of distributing a faculty annual salary over 9 or 12 months? It has been mentioned couple of times in this site that faculty could be able to distribute their salary over 9 or 12 months. What are advantages and disadvantages of each? And can faculty who distribute their salary over 12 months work during the summer? Do they get 1 month vacation and sick leave? Any effect on taxes, health insurance, and deduction for benefits? Does it affect the amount of summer salary from the university or a funding agency?
There may be some confusion here about the difference between 9 and 12 month appointments vs. arranging to have the salary for a 9 month appointment paid out over 12 months.  

In the U.S., the typical faculty appointment is a 9 month appointment.  Under this arrangement, a faculty member is required to be present on campus and teaching (and doing research and service) from September through May.  They typically do not have formal vacation or sick leave but in practice can stay home (and cancel classes) if sick.  They can also go on personal travel when classes are not in session.  Faculty don't have to formally request time off as long as they attend to their responsibilities.

In this system faculty do not have to work over the summers, although many faculty do teach summer school or work on research grants and earn additional salary over the summer.  Faculty will also often work on unfunded research projects over the summer.

Many faculty with 9 month appointments choose to have their annual salary paid out over 12 months.  This doesn't create any obligation to work over the summer and has no significant tax consequences, but it is more convenient for household budgeting since summer salary can be quite variable and unpredictable from year to year. 

For example, I just learned today that I will be able to earn a month of summer salary teaching a course that starts on June 8.  I'm still waiting to hear if some research funding for this summer will come through.  I've designed my personal budget so that I'm not dependent on summer salary, so I have my 9 month salary paid out over 12 months.

12 month appointments are quite common for academic administrators and some full time researchers.  These typically do accrue sick leave and paid vacation, which has some advantages but also has some disadvantages such as having to explicitly request time off.