Can using a breathing tube for a month damage my ability to breathe for myself?

Not really: If you are medically paralyzed, your breathing muscles (mostly the diaphragm) will be weakened. However, most patients are not completely paralyzed for that amount of time and you will be using your normal muscles to breathe. When the tube comes out after a month, you may feel a little weak but you should be able to breathe. The ICU will also help you get stronger before they take the tube out.

Probably not: The respiratory muscles can weaken if you are not using them enough, such as when you are on a ventilator. But those muscles also stengthen again. More likely, the reason you are on a ventilator probably includes lung injury, and if your lungs are not completely healed by the time you are extubated, you will notice trouble breathing on your own - not from the ventilator, but from the lung injury.