A little confused on how to properly integrate my sub

I mentioned it above, but the Elac Element EA101EQ is a great choice for a desktop integrated, I use one myself with a pair of Elac Uni-Fi UB5s and an SVS SB12-NSD for my computer audio.  It has digital and analog inputs, an asynchronous USB DAC, and an auto-blend function to automatically match your bookshelves to your subwoofer.   It's fine with 4ohm loads so the impedance swings of the LS50s won't bother it.

If you’re concerned about the amp section of an AVR, go for a modest Pre-pro and a stereo power amp of your choice/budget.

IIRC, NAD makes a higher end integrated with these features that’s pretty costly. DEQX and Lyngdorf make excellent products, but they’re even more expensive   The Elac recommendation above is probably the most cost effective solution that I’ve seen.

Okay, I didn't read all of the responses and someone may have caught this already. First, I personally think you're going the wrong way by selling the MF integrated in favor of the Outlaw. Especially considering how REL subs are intended to integrate. If you were to keep the MF, you would connect the sub's high-level (Speakon) jack to the speaker jacks on the amp. Leave the black wire hanging unless you get hum. If you do, ground the black wire to the chassis. You'd start near 12 o'clock with the x-over and volume on the sub and the phase at 0 deg. First, flip the phase switch on the sub back and forth until you figure out which setting produces more bass. Leave it that way. In my experience it's 0 degrees more often. Then adjust your crossover and volume settings until the sub blends smoothly with the speakers. You shouldn't really hear the sub per se, it should just extend the low end response of the speakers and improve sound staging etc. FYI, REL suggests their subs be placed in a corner if possible. The boundary reinforcement helps with extreme low frequencies but sub woofer placement is an important part of correct integration. You'll need to move and aim the sub so it sounds best and this will take some experimentation too. In my experience, even small adjustments make audible differences. Same with the speakers. The other thing I've noticed with REL subs is that they seem to me to like some break-in time and when they've been properly broken in they integrate more seamlessly. So do your best and don't be afraid to make minor adjustments over time.

A note on the difference between some of the other subs suggested compared to REL... Most sub woofers have significant group delay/impulse response time delay as a result of the amplifier and signal processing necessary to get small boxes to make big bass with extension down in to the mid-20HZ range or lower. This is not insignificant. JL Audio themselves will tell you that their subs have around 15 milliseconds of delay. This may not sound like a big deal, but it is. 1 millisecond is equal to about 1 foot so if a sub introduces 15 milliseconds of delay it's like placing your sub woofer 15 feet behind your speakers. REL handles design of amplifications differently and while their subs still introduce some delay, it's less significant. This is why the high-level connection works.
So if you do go with the Outlaw and you want to cross over and use the bass management, you can. Thought this is not recommended by REL. You WOULD use the .1/LFE input because it does not have a crossover in line. It's intended for connection to the LFE output on a AVR when used as a home theater sub and will work equally well in your application. If you do go this route, I would begin your setting up with the sub/speaker crossover point around 60 HZ and work from there. If the Outlaw has bass management in depth enough to include the ability to compensate for delay, with the REL I'd probably start with around 5-7' of delay and see how it works. You can always go up or down from there.

There has been a ton of pub in the last few years as audiophiles continue to embrace sub woofers as a viable part of a listening system. A lot of the pub is incomplete in my opinion. I think JL and many other manufacturers make good products but I don't think it's possible the achieve the level of seamless integration with them as you can with REL unless you have a way to time-align them at the crossover frequency. This requires DSP. Just adjusting phase on the sub does not do this. You cannot compensate for delay by adjusting phase on something that starts life 15 milliseconds behind your speakers. You must be able to delay your speakers so that they align with the sub. Many audiophiles are purists and adding DSP is sacrilege. For these people, in my opinion REL, or similar designs are the only way.

I've heard many REL subs in my years of designing home systems. Set up correctly, I think they are one of the single largest potential improvements in virtually any system and with almost any speaker. I've made JL subs work beautifully as well but with the addition of something like the McIntosh MEN220 which employs Lyngdorf RoomPerfect and the ability to make fine adjustments for delay. The last JL G213 took, if I remember correctly,  close  to 17' of compensation to time align with the Mac driven Sonus Faber main speakers. Being in the profession I have access to almost any product I could possibly want. I own two RELs. A T9i in my AV system and a S/3 in my small 2-channel system.

If your only source is your computer and you want to run "digital crossovers", the best approach is to run the digital xo inside the music software in your pc, then output to a multichannel DAC, and drive the sub/s on 2 channels and the LS50 on other 2 channels.

FWIW I started like this and now do 3-way active xo with a Lynx Hilo (DAC). 

I hadn't ever considered it, but Lewinski's idea strikes me as potentially very cost effective.  If I understand the proposal correctly, you'd only need to add the DAC and a multi channel power amp of your choice.  

However, the one caveat here is volume attenuation.  If it's done in the digital domain, you may find that there's an audible impact on SQ (particularly at lower listening levels).  As with all things audio, this is personal and you might also find that it has no impact on SQ.

If attenuation is controlled by the software, then you're effecting it in the digital domain.   If volume control is at the DAC, it could be in either (depending on the design of the DAC) the digital or analog section, but you'd hope it would be analog - in any event you can check the user's manual to find out.

I've never tried it, so I'm not taking a position one way or the other, just pointing out that - if the volume control is in the software - this is the "rationalist's" trade-off in this otherwise very rational solution.