Jack,
I think that you might have fun giving a mirrorless a new chance

With regards to the Nex-5 series, it's now at the 3rd generation (5, 5N and now 5R). The introduction of the 5N provided, according to users and reviewers, major improvements in all the aspects of the camera performance. The 5R is instead a 5N with a screen that can turn up to 180 degrees and wifi capabilities. Both the N and the R can mount an electronic viewfinder, so your liveview concern would not be an issue anymore.
Let's say that a 5R+EVF costs like a Nex6 (where the EVF is integrated) so it might be the case to get a 6 directly.
I got a 5N after the release of the 5R, so the price was heavily discounted. I might add the EVF when it'll get cheaper. In my case, having a dominant left eye, the integrated EVF would not be comfortable due to its position. When I tried the Nex7 my nose was constantly hitting on the LCD screen...
The external EVF is also tiltable, which is a nice option. The only thing is truly missing on a Nex5N (or R) is a mechanical knob for the shoot mode selection (like the one the Nex6 has). Otherwise the programmable functions make the 5 series pretty user friendly.
Regarding the Fuji, I think the X-E1 is a masterpiece of aesthetics. It has to be also truly good (unless you also want to shoot movies, a feature that even Fuji fan-boys consider badly implemented on the X series). Let's say that I envy the vintage look of the Fuji, despite the Gariz leather half case I picked for my Nex made it more traditional.
In terms of lenses, the native Fuji ones are regarded as better than the Sony counterparts, despite they're not stabilized and they cost more. I've never tried any, so I cannot tell by personal experience. I guess that in my unexperienced hands the 2 different systems would perform just the same... OTOH, the Sony system has opened to third party produced native lenses, so you can find affordable (but apparently super good ones) lenses from Sigma, Rokinon and SLR Magic (the latter, together with Zeiss, now produce both for Sony and for Fuji).
Anyway, this Sony vs Fuji competition mirrors(less) the eternal fight between Nikon and Canon among the D-SLRs... there's never be a winner.
Yessir. EF compatible (with an adapter).
I owned a Sony NEX5. Late last year before leaving for Spain, at the very last minute I decided to leave my D700 because of the weight. I picked up the little Sony at the HK airport during the short layover. It's small, light, easy to use (nice manual control layout), takes good pictures. My only problem with cameras of this sort is having to rely on live view 100% of the time. When it is bright out, taking shots is a crapshoot since you have hardly any idea what the sensor is seeing. The 5 also doesn't have a built in flash. I gave it to my daughter when we got back and she absolutely loves it. I'm eyeing a Fuji. I was eyeing an M9 but when I did the math, I said I'd rather buy an interconnect. Just kidding Frantz! LOL!!!!!!!!!!!!!!!!!!