Well considering it clearly says that in the event of a deep wound or puncture wound you should seek medical advice (it's there way of saying not to use the product on those types of wounds) and a piercing is a puncture wound. I would suggest you don't use it. Neosporin contains neomyacin and this needs the air circulation to work effectively and a lot of people are sensitive to Neomyacin. If you are washing the piercing daily using a good unscented liquid antibacterial soap and rinsing with lots of warm running water while moving the jewellery there should be no reason to use Neosporin on your piercing.
Neosporin and Polysporin are never to be used on any body piercing, read the back of the tube. It clearly states not to be used on puncture wounds, a body piercing is a puncture wound. Neosporin contains Neomyacin and Polysporin contains Polymyacin both of those ingredients will cause issues with puncture wounds or body piercings. Most people will react to these ingredients in an adverse manner when it's introduced into the body. These are topical medications for surface injuries only.
I would recommend you don't use Neosporin, Polysporin or anything like that on a ear piercing. Soap and water will do just fine. These products are not to be used on puncture wounds (read the tube). If the piercing is infected washing with diluted antibacterial soap and irrigating with water twice a day will make a big difference in about 24 to 48 hours.
The tube of Neosporin doesn't say to use it on puncture wounds, so you don't use it on puncture wounds.
You shouldn't because it doesn't let your piercing breathe
You can, but you are better off using a product that will treat infection, such as Neosporin.
No, you are only supposed to use soap to clean the piercing. That includes no peroxide, alcohol, etc.
Yes Neosporin is a good medication to use if your Beardie has an external sore or injury
No. You should never under any circumstances put anything on your piercing such as Peroxide, Alcohol, Neosporin or Bactine. Peroxide, Alcohol and Bactine are very drying to your piercing and the surrounding skin, and will just make the problem worse. Neosporin dries inside your piercing and clogs it, causing a flare up. Regardless of what people may say, the only thing that will fully cure an infected piercing are antibiotics. Other products may mask the symptoms, leading you to believe the infection is gone, but in all reality it isnt. If you think your piercing is infected, see your doctor for antibiotics.
Is the cut related to the piercing or is it just from a scratch in that area? If its unrelated to the piercing, you can put some neosporin or any other antibacterial cream on it. Make sure its the cream and not the ointment because the ointment is too thick and can stop the piercing from breathing if the cut is close to the piercing. If the piercing hole looks like a slit because of the cut or migration, go see your piercer. You might need a custom bar or to remove the piercing and get it done again later.
Try following the aftercare instructions you were given by your piercer first, I think you will find you should not be using anything not listed in the instructions.
You can't use the ointment but you can use the cream. The ointment is too thick and suffocates the piercing. You really don't need it though. Unless you scratch it with jewelery or something. Natural sea salt and water is fine for cleaning.
You really don't have too. If its infected and your doctor tells you to use an antibacerial cream, then you should. But otherwise, it'll heal just fine without it. Also, if you do have to use it in the case of infection, use the cream, not the ointment. The ointment is super thick so it essentially suffocates the piercing and it collects dirt/dust. The cream rubs in so its suitable for a piercing :) **The best thing to do for your piercing is a good saline soak (AKA, salt water). If your piercing is giving you difficulty, then you should soak it for 5-10 minutes in a hot salt water solution. Even better than salt water though is emu oil. It's basically the miracle "ointment" for body piercings. Though neosporin does have a history of being used on body piercings, I advise against it. It can cause even more irritation in your piercing.