Are all USB "audio adapters" DAC's? Can USB ports be used for analog output?
Can USB ports output analog audio to these small adapters?


No. Well, there are cell phones with USB ports that have multiple uses, when they can output analog audio in dock mode, but that's not the case with 99% of computers.


  Or do all USB to jack adapters have a DAC?


Either external or internal to it's USB IC, yes.


  Is my DAC/Adapter outputting noise because there is noise on the USB power supply from the motherboard?


Most likely. Same reason the HDMI does not. One, HDMI cables are designed to avoid ground loops. Two, the TV has it's own power supply, so only the data from the HDMI connection is used. Three, the TV probably has additional audio filtering, as size is not really a concern.


  Would a new bigger DAC with it's own power supply fix this?


Maybe. Theoretically it should, but in practice, YMMV.