Can I connect other bluetooth devices such as headphones or ear buds and listen to normal broadcast audio when not going through the ROKU APP?
It depends on which model number Ultra you have (Settings / System / About will tell you).
Some of the newer Roku devices with the latest chips now have the capability of sending audio to Bluetooth devices directly. See this Roku article for details and list of Roku devices supporting this: https://support.roku.com/article/12210854415383
Aside from the specific Roku models mentioned in that article, you can not directly send audio from the Roku to other devices via Bluetooth. For this you must use the Roku Remote app on a phone or tablet, and connect the Bluetooth audio device to the phone/tablet.
So, as I understand it, if my ROKU ULTRA is a model 4800 or 4802, it will pair with my bluetooth ear buds or headphones but, can I listen to a cable broadcast or will it only play audio on my paired device through the ROKU APP?
Per the article cited above, Ultras 4800 and 4802 can connect directly to Bluetooth earphones and speakers for private listening. You could still do your private listening via the Roku app if you prefer, same as if you were not using one of the Bluetooth capable devices.
Either way, you would be able to listen to the audio for anything playing on the Roku.
See the cited article for full details.