Can a month old infant fly? Are there any health related issues prevent that or is it ok? Any restrictions from airlines I may face?
There are recommendations from Pediatricians that state the child should be 2 to 3 months old before traveling on an airplane. There is ample evidence in the same link that it could be safe enough for even a younger child to travel and a confirmation of this from Mayo Clinic. However, it is up to the individual airline to set the policy on how young is too young.
For example, United won't accept
But US Air doesn't impose such a restriction.
As far as health issues are concerned you should be concerned with 2 major ones change of pressure during takeoff and landing, babies don't know how to deal with it so you will get crying. And second being that child's immune system is not as developed as the adult one, so he/she might catch something that someone else is carrying that you might be immune to.
The last question will depend on your airline. But for instance, United Airlines says infants younger than seven days old may not travel, implying that infants older than seven days are generally permitted.