Could I have a gum disease?!

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The redder the gums the more inflamed they are. Healthy gums are pale pink. You likely don't have full blown gum disease with bone loss, but probably gingivitis. This can lead to gum disease though and it's important to take care of it.
You want to reduce the bacteria load in your mouth, one is with fixing that broken tooth and the other one is with good home care which includes brushing twice a day, flossing once and rinse once. You want to make sure you brush your gums as well, ignore the bleeding if there is any and your gums will get better.

If you haven't had a cleaning by a dental hygienist in a while you may need to do that first before your gums get better. There's also certain medication, bad fillings/crowns or mouth breathing that can affect your gums. Ask your hygienist about it, she'll know.