Yes, you can overtrain from doing an excessive amount of exercise even cardio. Overtraining "is a physical, behavioral, and emotional condition that occurs when the volume and intensity of an individual's exercise exceeds their recovery capacity". You can get injured and you can show signs of emotional and behavioral changes from overtraining.
Everybody is different so there is no one-size-fits-all rule, but there are general guidelines to help people avoid overtraining during cardio: See previous question, Is there a rule of thumb for setting running goals
"Too much too soon" is a common phrase to describe why people got injured. The Cool Running website describes phases of physical shape you should be in as a disclaimer before attempting their rigorous training program (shown is half-marathon training phases):
For runners who currently run 15 to 25 miles per week and expect to run the half marathon in about 2 hours.
For runners who curently run 25 to 50 miles per week and expect to run the half marathon in under 1:45:00.
For runners who currently run 40 to 60 miles per week and expect to run the half marathon in under 1:30:00.
For runners who currently run over 60 miles per week and expect to run the half marathon in under 1:20:00.
Cardio is all about making the muscles you use more efficient. Say you run 3 miles a day, your body gets into a routine. Your muscles get trained to keep running that distance and naturally your endurance will increase and running that distance will feel "easier" on you. Your heart rate isn't as high as when you started training; you're body is managing oxygen better. As you push yourself, increasing your miles, you push your endurance to also increase with you.
Yes, more cardio training will make your lungs and heart operate more efficiently.
ASIDE: I know it wasn't asked but it is important to note that changing your regimen to a majority of cardio will change your muscle composition to slow-twitch (long and thin) muscles which are more efficient for endurance. If you want to target your fast-twitch muscles you got from weight lifting, you should be doing high-intensity interval training (HIIT) in conjunction with your strength training.