Airfares are typically valid based on the flight dates, including the return flight unless it's an open-ended ticket or something like that.

The time of booking is an entirely different parameter from the dates of the flights. Obviously by catching a good promotion, you can save money by booking at the right time, although this is more of a factor for off-peak season booking. (i.e., Europe in the winter.)

Peak (highest airfare) season can vary by route. For example, the toughest ticket to a tropical destination might be Dec-Apr (with wide-open availability in the summer) whereas the toughest ticket to the UK is Jun-Sep. I don't know about LAX-Asia. Northern hemisphere summer would be a good guess for peak season to most places.