Cusp on upper molar broke off below gum; two-thirds of tooth remain. What are the chances I will get an abscess or other gum infection?

High. Broken tooth easily infected which leads to abscess. Gum around fractured area unstable. Please see your dentist before your problem gets out of hand, which can happen quickly.
Hard to say... Did the tooth have root canal therapy? Is the tooth vital? Is the fracture through the furcation? 1/3 of the tooth is now missing, which means it needs to be restored. You'll need to see a local dentist to have the tooth evaluated and have the above questions answered. Only then will you know what your options are, what it will cost, and if the tooth will get infected.