Another title is: General Manager.
If you are addressing someone as the president then yes you do capitalize it.
President Director is not a common title in the United States, but it is not unheard of. CEO is a more common title for the top executive with a company.
This depends on the company and the full title if any.
mr. president
CEO
Yes.
When used as part of a title, yes, e.g. President Obama. When used as a common noun, no, e.g. "The president [of the company] had a meeting with the board yesterday".
Chief Executive of the United States
Commander and Chief of the United States
Yes you can but you have to ask the company first
Yes they can I work for the company
If two people were on title and one of them took out another mortgage on the property then the title company is at falt. In this case you have to talk to the title company and perhapes a real estate attorney.