Is it a bad thing to put my MacBook Air 13" directly on my bed not raised on a flat surface? I need to know if any damage will be caused because of it not being able to breathe.

Here is a "copy and paste" from user's guide.

To operate the computer safely and reduce the possibility of heat-related injuries, follow these guidelines:

 Set up your MacBook Pro on a stable work surface that allows for adequate air circulation under and around the computer.

 Do not operate your MacBook Pro on a pillow, blanket, or other soft material, because the material can block the airflow vents.

 Never place anything over the keyboard when operating your MacBook Pro.

 Do not push objects into the ventilation openings.

 If your MacBook Pro is on your lap and gets uncomfortably warm, remove it from
your lap and place it on a stable work surface.

Is it a bad thing to put my MacBook Air 13" directly on my bed not raised on a flat surface?

Yes, yes, and Yes

Considerations for the long life of your Macbook Air. Handling and environmental care tips

Don’t use your MacBook Air on the beadspread or pillow

Do not place your MacBook on a pillow or other soft material when it's powered on, as the material can block the airflow vents (in particular, the rear vents) and cause the computer to overheat. Never place anything over your keyboard when operating in closed-lid (clamshell) mode. This can cause your computer to cycle on and off, which might create excessive heat and drain your battery. Use a lapdesk if you want or plan on using your Macbook on your bed or couch, or in your lap.

Use of your notebook on a mattress or pillow blocks your notebook from circulating air as it needs to, and can lead to overheating rapidly

Lap desk for use with your notebook on the bed, your lap, or couch. Also very useful for removing wrist strain