Laptop to 4K Monitor

No, it's not possible. Both the on-board Intel HD4000 graphics and the Nvidia NVS 5400M have the same maximum resolution of 2560x1600.

From the Intel spec sheet linked above, newer 4th generation Intel GPUs do support 4k (3840x2160 and 4096x2304) at 60Hz.

Yes, it is certainly possible.

All you need is an Active adapter like Accell B086B-007B.

As of mid-2014, it seems like there are a lot of 15 USD adapters that are advertised to support Ultra HD resolutions, but which do not appear to be as active as the active 25 USD ones by Accell, and none of those other adapters seem to be able to support Ultra HD on most of the laptops.

I've personally tried connecting ThinkPad X230 w/ Ivy-Bridge to Seiki SE39UY04 with this Accell B086B-007B-2, and it works -- I was able to get 3840x2160@30Hz, even though the spec for my X230 claims Maximum external resolution: 2560x1600@60Hz. I've even tried it with a late-2008 MacBook, and to my surprise, even that combination worked!

If you search around, you'll find lots of surprised people who likewise found out that their very-very old mini-DP hardware works with Ultra HD just fine.