What about the subwoofer?

You never want to use a Y-cable to do this.   What happens is the voltage coming out of the AVR will feed back into the integrated amp and can damage those audio circuits.  If this is the direction you are going in, what you really need is a mixer (which is a completely different type of audio circuit).  You can find cheap RCA only audio mixers from pro audio companies (like Sweetware).   Balanced XLR mixers are also available but are more expensive.    You can also explore passive mixers (without an active opamp type circuit), but these can have their own problems since they drop the voltage level of the audio signal.

Otherwise, the options are sending audio cable to subwoofer for sub crossover before being connected to the main left/right amplifier.   Or doing a high level tap of the main amplifier speaker connectors to be fed to the subwoofer (this only provide additional bass support to the main speakers).

auxinput

I appreciate the response. As you see, my interest is to be able to use the subwoofer in connection with the front speakers for stereo listening after installing the integrated amp, running L&R pre outs to it and running my streamer and dac to the preamp instead of to the AVR, as currently. In this configuration the subwoofer would be directly connected to the integrated. This was all based on the understanding that with the separate setup the AVR would be completely out of the signal path for stereo listening and therefore would not have to be turned on.

However, setting up the sub that way would not allow it to be used at all with the AVR, thus not available when viewing movies. That is why I questioned the possibility of using the Y connector.

If the sub remained connected to the AVR and the AVR were powered on along with the integrated when stereo listening would the sub still function? This assumes, I suppose, that the AVR is set to L,R, and Sub, as it is currently when I play only stereo thru the AVR  

Well, you are using the integrated amp as an "amplifier" left/right outputs of your AVR.  So you would still be able to take a high-level tap off the speaker outputs of your integrated and use that for the subwoofer.   Like I said, the left/right would have the full-range sound and would not be crossed over.   The subwoofer would just add any low bass that your left/right would not be able to produce.  Not the best solution, but the only solution you would be able to setup.

Seems that the OP is adding an additional amp between the subwoofer signal and the subwoofer, then trying to bypass the added amp like there was a “subwoofer bypass” feature/option

The reason a preamp processor and separate power amplifiers are suggested for hifi applications over a AVR is...

- The preamp processor (no built in amplication) is usually a ground up designed flagship product from most brands with the best preamp sections, dac implementation, streaming function, dedicated power supply, etc. It can still be relatively affordable because it gets subsidized by the higher end HT guys (markets of scale). 

- You can connect it to the best power amplifiers (depending on your budget). The primary weak link in an AVR are the compromised power amp sections crammed into the same box, injecting noise everywhere...downgrading the upstream components and everything in the box, brings everything down.

By using some dumb integrated amp, you lose very useful features like bass management, peq, etc (that the prepro came with). You will find it very difficult to integrate subs correctly and so on with a purist piece. By deploying a high quality preamp processor paired with hifi power amplifiers, you can achieve very good results when running it in stereo mode as well. 

Get a prepro from a good brand and use the best power amps you can afford for the front left, front right and center channels. You can get by with some power amp compromise for the surrounds and height channels.

Refer to my room 2 system for a ballpark on what can quality as both a hifi 2 channel and multichannel hybrid system.

(Sky’s the limit if you have less wallet restrictions, of course).