What is having an anesthetic injection for a tooth filling like?

Like a fat lip/cheek. The anesthetic injection will feel like a pinch, depending on where in the mouth you are getting the injection. Some areas in the mouth, usually in the upper front area may be more sensitive than other areas. Usually your dentist will apply a topical numbing gel prior to the injection. After the injection, the numb feeling will feel like a fat lip or cheek.
Not too bad. It will feel like a small pinch when getting it at first, then a filling up feeling in the gums, then like your face is asleep. Similar to when your leg or arm falls asleep and you have to move it and shake it to get the feeling back. Important to remember you are numb, and be careful not to bite yourself.
Not bad. Most of the time a topical gel is applied to the injection site first. This is usually a flavored gel like mint. It's left in place for about a min. To numb the gum tissue. When the needle is inserted there may be a little pinch and as the medicine is delivered some additional pressure may be observed. Usually quick and relatively painless if done correctly.