(The name in question is Koenig. Pronounced "Kay - nig". Not "Ko - ning")
Where does your name originate from? I can't think of any language where "oe" is pronounced "ay".
...and I've never understood the point of placing punctuation marks within the quotation marks, but I should probably conform...
Indeed. Koenig can't be pronounced Kaynig
Regardless if you're German, Swedish, or whatever...
As for the punctuation, American typesetters put . and , inside the " due to the physical size of the printing things; it was more practical as they were so narrow. That's all... English people are more sensible and place the punctuation where it belongs.
Whether or not punctuation should go inside or outside the quotation marks depends on whether it belongs in the sentence within the quotes, or the sentence outside the quotes. In other words, if the sentence within the quotes ends in a full stop or whatever, it goes within the quotes, and you put nothing after it. If the phrase in the quotes does not end in a punctuation mark, then you can put punctuation at the end of the outside sentence. For example:
The man cried out, "Help me!"
This has punctuation inside the quotes, as opposed to this sentence:
The man said he came from some place he called "BakaBT". I wasn't sure I believed him.
This one has punctuation outside the quotes, because what is in the quotes does not end in a full stop, exclamation mark, or whatever. Contrast with this:
The man said "I come from a place called BakaBT." I wasn't sure I believed him.
This has punctuation at the end of the quoted phrase, so you can't put any following the quotes.
Yay for grammar. It's such fun.