I would use a chrombook then becuase that is your best way to get good battery life, try the acer chrombook 15 the best model has a 15 inch 1920*1080 screen and costs only 340 or so http://us.acer.com/ac/en/US/content/model/NX.MUNAA.003
I would personally recommend a business laptop ideally a ultrabook as from my experience they just have less issues in general, the other options would be either to buy a Chromebook, or to get either a surface pro 3 or surface 3. Quick question though what screen size do you want? expected...
Nope its deffintly your hard drive but if you want to confirm it donload this http://crystalmark.info/software/CrystalDiskInfo/index-e.html on a working computer plug in your hard drive, and then run it on the disk
What is more likely Is two issues, 1. That enourmise data rates required to display 4k, and second the added workload of having to upscale to 3840*2160, a trick is try running stuff at 1080p as that scales nativly to 3840*2160 since it is twice the width and height of 1080p
one of two things first go into BIOS and check that your hard drive is in the boot order. if not either your motherboard, or more likly your hard drive is FUBAR
check out the eurocom m5 pro http://www.eurocom.com/ec/configure(1,257,0)ec really good deal though not quite as small but still good and can acomindate dual hdd and msata drives
its similar to the surface pro 3 in that its reall a tablet running windows with a (realitivly) well designed keyboard dock, however its process is much weaker then either the surface pro or lenovo yoga
couple of questions first what screen size do you want? second what batery life do you need? and third do you need a laptop as you would be beter served by a latop+ desktop