Will the this lens fit the NIKON D5200?

Will the this lens fit the NIKON D5200?
Added (1). http://www.camerajungle.co.uk/item/205825/nikon-1-nikkor-vr-10-30mm-f35-56-nikon-1-mount-zoom-lens.html

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What lens do I get first?

So I have got a Nikon D3300, I love photography as a hobby and want to take it to another level. What lens should I get next. At the moment I have got the 18-55 kit lens which really doesn't zoom far enough.

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Help picking a mid-level camera?

I am looking for a digital camera I can purchase on Amazon (it's a gift card) for at or below $150. In order of most important:
#1 price - if I HAVE to I could save up to a limit of $375…$100-$200 is better though
#2 compatibility w/a shutter remote (would appreciate if you know a good one too)
#3 compatible w/most tripods (I already have 2)
#4 anti-shake/stabilizer
#5 resolution/quality of still image
#6 SIGNIFICANT superiority to smartphones & tablets
#7 BONUS video quality - including ability to pick up sound from a few feet away

I need a nonbiased person ideally someone who can tell me the REAL difference between "new", "used", & "refurbished"… I've always gone for brands I recognized & only brand new but am trying to keep an open mind

I got a chance to see the Nikon S1 being used and was impressed tho don't know if it will fulfill my remote shutter accessory requirement (I don't expect is to be included but MUST be compatible) but is DOES have the option to switch lenses like my old one, the only other downside I could see is, according to this Amazon listing for this:
Nikon 1 S1 10.1MP Red Digital Camera with 11-27.5mm Lens (Certified Refurbished) https://www.amazon.com/dp/B00THM31D0/ref=cm_sw_r_awd_FqOAvb03N097S
which an identical (though refurbished) cam to the mom's cam is that it is only 10 megapixels and I was hoping for at least 12 if not even as high as 16.

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