How to test out all functions of a hero alone

No. All of the heroes in Overwatch are (and will be, in the case of future additions) unlocked and at full power from the start. The only things you unlock through either time or money are cosmetics, such as skins, sprays, or victory poses.

What you are looking for is the Custom Game option in the Play menu. You can choose the map, the map rotation, the difficulty and amount of AI players, etc. This should allow you to play privately or with friends in order to increase your knowledge of maps and heroes.

http://overwatch.wikia.com/wiki/Custom_Game

Edit: As to how the youtubers and others fly around the map, they use the custom game menu, and when viewing the team lineups page, they right click on their name and swap teams to the Spectator team. Then they start the match. This video shows how it can be done at around the 10:00 mark: