As dutchraptor said you can't go from a dream into another, deeper, dream, but you can experience what are called false awakenings, though you have no control over them. They're basically waking up into another dream, so unlike inception going further down into dreams, you're coming up into dreams, though time perception is a different story. Some people do what's called time dilation where you make time go by slower in a dream so that if you take a 10 minute nap you could have a dream that lasted, say, 20 minutes. Some claim to have done this for years intentionally but I find that about 5x normal time is where I draw the line, but you're free to make your own conclusions.

As for the breathing problem just focus your attention elsewhere. Once you become lucid focus on your RC, then your stabilization, then whatever you're doing so you don't think about your breathing. When you're doing something in waking life like playing a sport you aren't really thinking about your breathing since you have other, external stimuli, but when you're laying still trying to fall asleep it becomes something you control. Just use the same concept in a dream and you should be fine.