Is it ok if my 1 month old falls asleep in her rocking swing at night?

I'd advise against. Infants are not aware of sleep until well into the 2nd yr.They drift into sleep like u do listening to a sermon that goes too long. They need a comfortable sleep zone;enough light to be visually recognize the setting, simple background sound and a mattress to be lain on.Reading to the tired kid from a distance helps them settle.Kids rocked to sleeep have 3-4x sleep disturbences for the first 4 yr.
No. Depending on the type of swing you are using the danger is if she slips down and is strangled by the belt on the swing. Also reflux and breathing problems can occur. The recent guidelines (http://tinyurl.Com/72amzj4) on baby sleep are very strict (firm mattress, no bumper pads, no quilts etc.). Link to the full sleep guideline http://pediatrics.Aappublications.Org/content/128/5/e1341.Full.
Sure. Once she does, just gently get her out and put her in her crib. I wouldn't let her sleep the night in the swing.