Typically, the summer months are the business for moving and therefore the most difficult. If you can move in a non summer month, it should be cheaper and easier.
This will vary depending on the mover but typically they will calculate the amount of hours required, the number of workers, and the overall quantity of belongings that need to be moved to arrive at a all in number.
When moving, be sure to process an address change with the USPS and the other relevant government agencies so you receive all of your mail.
The earlier the better, but 3-4 weeks prior to the move is recommended.
This depends on what you request. Almost all movers will pack up your home if you prefer, though there will be additional charges related to packing everything up.