Generally a bung is not going to affect your beer. People lose their bungs in the primary all the time.
Oxygen is bad AFTER primary. Before and very early on in the primary oxygen is used by the yeast for reproduction and for cell health, and is usually very important for good beer quality (liquid yeast. Dry yeast is reportedly fine without oxygenation as it's sterol reserves are built up during the drying process, but that's not the point to this topic).
Once the yeast has used up the available oxygen it really kicks into alcohol and CO2 production and the real action starts. Now once the yeast has taken the oxygen out of the wort/beer the beer is safe. Once the sugars have been eaten, the yeast will go dormant and oxygen introduced will remain in the beer and react with some of the other compounds to create oxidation characteristics.
Remember that it's not the oxygen itself that is noticeable, it's the result of other compounds reacting with the oxygen.
So basically oxygen is bad after the yeast have used all of the oxygen up in the primary.
Avoid splashing your beer when racking. Avoid agitation or splashing when bottling.
Purge air from bottles or kegs if you can before you fill them. Use O2 caps if you can.
Heat accelerates oxidation. Once your beer is carbed, keeping them cool will really help slow down any potential oxidation.
Oxidation is harder to control with bottles because if you keg you already have CO2 and a means of easily purging the air from a keg before filling, and for purging AFTER the keg is filled.
Some beer styles almost require some level of oxidation, but they are usually very strong beers that have been aged. It's almost never a desired trait.