jedelen,
You can access the vast amounts of aircraft by flying Quick missions, or by creating a mission and selection various aircraft to be available.
Otherwise, you can access the entire list by playing multi-player in "Dogfight" mode.
As far as multi-player goes, you have a number of options, such as Hyperlobby and personal hosting among friends, etc.
The online manual/handbook (PDF) was chosen because of the size of the book, if it were to be published. You can either look up a specific aircraft profile from the handbook PDF, or you can go to the main game menu and select "View Objects". From there, select "Aircraft", and then select "Country" from the pulldown menu. That will then show all the aircraft in the list below and clicking on a specific aircraft type in that list will display information about that aircraft in the main window to the right. If you wish to see that aircraft's image, click the "View Object" button below the main window, and it will render a 3D image of that aircraft in the main window.
You can return to the text by clicking the same button.