When you use a dynamic text box, you have two options: either you insert (part) of the font into your SWF, or you use the device’s fonts. If you insert, then the actual character characters will be embedded in your SWF; if you do not, then you include only the font name - and if the OS does not have fonts for that name, it will choose the default instead.
From your problems with the field that you copied from another file, it looks like you might have missed the "Paste" settings. Look at the button that says “Insert Settings” or the like in the Property Inspector.
If you decide Embed, then you are guaranteed that your text will be displayed in Arial on all platforms. However, this is only true for the characters you insert. If you insert only uppercase letters and then enter the text “Hello”, all you see on the screen is “H”. (Be careful with the attachment of the entire font - for full-format Unicode fonts that will have several megabytes, since they include Japanese and Chinese, etc.)
If you decide not to embed, then you should probably use one of the “devices” of fonts listed first in the font menu: _sans, _serif, _typewriter , to avoid problems you may have. In almost all cases, they will be translated into Arial, Times and Courier on PC and similar fonts on Mac.