Okay, so first off:
AMD is launching their mid range cards first, and Nvidia is launching their (higher) end cards first. AMD is definitely planning on releasing higher end cards, and Nvidia lower end cards. I would dare say, given the tiny size of the 1080, Nvidia probably plans to go even bigger soon, frankly.
Next, rumor says the mobile versions of Nvidias cards will be their desktop versions. Since Nvidias cards are power efficient, this makes sense. This could be true for AMD as well.
We simply don't actually know how they will stack up performance wise, AMD has not really let reviewers play with their RX 480 cards yet. Given the recent benchmarks of new games using directx 11 technologies, and the manufacturing technology jump, I actually expect both AMD and Nvidia to perform very similarly, probably with a cost advantage on amd.
In terms of launch dates, thats pretty up in the air. I expect something by August of this year, but I don't know what the market will look like. By the end of the year, we will definitely have new laptop cards from both manufacturers.
Lastly, your standards are very low. My 4 year old desktop with a gtx 680 can play most games at high settings just fine, and my lower end laptop ($1000) with a gtx 860m can still play most games at medium (though, I did have to push it to 720p at low to get consistent 60fps in dark souls 3). You can definitely buy a laptop right now that will give you medium settings or better for the next 4 years at the $2000 price point. That said, you should totally wait and get the new graphics cards because the improvement is huge, but don't worry about getting a machine to play your games at low to medium settings.