Having read all the above comments, and added a few by myself, I would, after having handled both the E-M10, and the GX8, that I'd jump on the E-M10 II offer, and buy the 1.7/20 Pan lens, and still be within budget. The kit lens isn't fast, but collapses to a mere nothing, and that Pan lens is so sweet, and fast! The camera has a nice viewfinder, flash, and seems rugged.
And I've heard no horror tales, either!
I myself love the Nikon 1 system, like the V2, which normally comes with the kit zoom (10-30) and there is an excellent 18.5/1.8 (equals a 25 on a m43 camera) that is still amazingly cheap. Add a 30-110 (which in m43 terms is a 40-150), which is an outstanding lens for the money! I'd say you'll get all that, and still be within budget, if you buy a used V2. Sometimes new ones can be found at about $250, but have not seen those prices lately.
The V2 can be found new on eBay, Amazon, and similar places, but it is no longer in production. The V1 lacks flash, so not quite according to your specs. Nice video on both!
Later versions either lack viewfinder, or flash, on the V3 you can switch between the two, but not use them at the same time!
The V2 is very compact, takes very little room, sweet little camera!
A few shots taken with just that little camera:
All the mammals captive, all the birds wild! Almost all shot free-hand.
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Tord_2 (at) photographer (dot) net
Mostly Nikon V1, V2, J5, & D600, user