Yes, it appears that the Tascam DA-3000 does have what you would call a "streaming mode". You can enable a mode called "ADDA DIRECT" which basically takes any analog audio coming in on one of the inputs and converts and then output that on the analog and S/PDIF digital outputs (it basically becomes a live AD/DA converter). However, you will still need a digital audio interface to connect it to a computer for recording (such as Focusrite, MOTU, RME, etc.). This sounds like it defeats your entire purpose of getting the Tascam in the first place.
The USB inputs on the front of the Tascam do not connect to a computer. They are meant for a keyboard and a USB drive to record music onto.
Sounds like a convenient way to record analog sources, but you still need a USB drive or a Compact Flash memory card to copy those files to computer and rename them. It does look like this Tascam will do DSD recordings if you so desire.