VCheng has perfectly answered your query. However I would also like to add my 2 cents.
Firstly, for the budget of Nikon D5200 or D5300 you can easily find a used D7000 with 18-105. Condition being that the unit should be perfect. It will serve you better than the D5200/D5300 since it carries some higher end capabilities and not to forget the capability of built-in focus motors (lens upgrades cost less). If you want your investment to be a brand new one then I would recommend a D5200 with a 18-105 rather than the standard 18-55. And when you have the budget later on for a small lens addition, go for a 50mm f/1.8G. More focal length with a fixed aperture will give you better bokeh plus the quality on the 50mm is better than the 35mm. 50mm also supports FX bodies so future is safe.
Secondly, if the intention to purchase is purely learning photography then by all means get a DSLR. However if the purpose is time-kill with quality pictures wherever you go, then I would either recommend a mirror-less or good high end Point & Shoot. 2 reasons, the first being convenience and the second being no hassle of switching modes (even high-end P&S carry manual/aperture/shutter modes with decent sensor size)
Thirdly, Option 1 is NOT feasible since the lens they provide you is the D version which lacks auto-focus capability. Since the D5200 body does not have a built-in focus motor, it will all the time require you to manually adjust the focus every time you want to take a picture, by rotating the focus dial on the lens.
Fourthly, I used D5100 extensively and it still is at par with the latest offerings in the D5000 series. The quality is the same, just the extra features for flashy kids are there. One would rather spend the extra money on lenses.
Lastly, I would propose that if you just want to learn, start from D5100 with a good lens or a D7000 with its kit lens. If you have the concern of buying new equipment only then get the D5200 with a better lens i.e. 18-105. (As per my experience the 18-55 sells for 8k and the 18-105 comes for 28k, this experience is quite old though). If you just want to pass time and enjoy taking pictures (carry your camera everywhere) get a good high-end Point and Shoot.
Apologies for my response being quite long (might be confusing). I hope I did justice to your query. It's been a long time since someone pinched the photography insect here at PW.