I don't think the DGX-660 has a built-in USB audio interface (or "Audio Over USB").
It can read and write audio to its USB TO DEVICE port, meaning it can record audio on a connected USB flash drive and play back audio files from a flash drive.
But as far as I know it cannot transmit and receive audio data via its USB TO HOST port.
At least some of Yamaha's synths have that feature, such as the MOXF6/MOXF8 and MX49/MX61/MX88.
But I think the only keyboards with that feature are the PSR-E363/PSR-EW300, the PSR-E453/PSR/EW400, and the PSR-E463/PSR-EW410.
You can tell if a particular model has a built-in USB audio interface by looking at the Specs tab on the keyboard's product page at Yamaha's web site. If the model has Audio Over USB, then the line where "USB TO HOST" is listed (under "Connectivity") will say something like "44.1kHz, 16bit, stereo"; otherwise, it will simply say "Yes."
I haven't checked whether any of Yamaha's P-series digital pianos have Audio Over USB.
If you want an 88-note keyboard with weighted piano-style keys that has Audio Over USB, you'll probably need to get either an MX88 or a MOXF8.
Otherwise, if a 76-note keyboard with unweighted synth-style keys would be satisfactory, look at the PSR-EW300, PSR-EW400, or PSR-EW410.
And if a 61-note keyboard would be sufficient, look at the PSR-E363, PSR-E453, PSR-E463, MX61, or MOXF6.
Now, as to your actual questions...
Yes, if you get a keyboard that has a built-in USB audio interface, it will show up on the computer as an available audio source and audio output. The computer will be able to receive audio from the keyboard over USB, and you can have the computer send audio to the keyboard over USB.
In Windows 10, you can easily select the keyboard as the computer's audio output device by clicking on the little speaker icon in the system tray (or the right-hand corner of the task bar). A slider should pop up to let you adjust the volume of the audio, but listed just above the slider should be the audio output device. If multiple devices are available, you should see a downward-pointing arrowhead to the right of the current audio output device. Click on that and the list of currently-available devices should drop down. Just choose the Yamaha keyboard and-- if it is equipped with speakers-- the keyboard's speakers will play any audio that would otherwise go to the computer's speakers.
Of course, that's for the computer as a whole. You might still need to select the keyboard as the audio device in whatever apps you are running-- assuming they have settings that let you choose between different audio output devices. But in general, if an app is sending its audio output to the computer's speakers, then you can redirect it to the keyboard's speakers as described above.
The connection should be stable and reliable-- if the USB cable is in good shape, that is.
I'm afraid I can't answer your question about Android tablets. For one thing, Android tablets can vary greatly as far as what types of connections they can support, so it's conceivable you might not be able to connect a keyboard to your particular Android tablet via a USB cable.