AR10 vs AR 15
The "Original" AR 10 wasn't Dutch, it was American. It was designed By Eugene Stoner, and was not at all like the AR10 of today. Aside from the visual differences, it was piston operated & 7.62mm (.308) It lost out on the army contract to the M14, largely because of politics. Several years later the army came back to the idea, but they wanted it lighter and in a smaller calibre so that the grunts could carry more bullets. After a complete redesign, we got the M16.
So today's AR15's are the children of the original AR10. HOWEVER, today's AR10's are actually just a big AR15 as opposed to the original AR10 design.
Knight's armament started looking at bringing back the AR10 concept a while ago for the SEALs, and then Armalite brought it back commercially, using modified M14 mags. To take down an AR10 and an AR15 side by side, you would see that the parts are almost identical, just bigger on the .308 version. That's why the receiver is longer, because the 7.62 round & mag are much longer. AS a side affect of Armalite using modified M14 mags for their version of the AR10, it's mags aren't interchangeable with other "AR10 type rifles," for example DPMS.
By the way, the rifle in your picture is an Armalite, with the modified M14 mag.
Any AR10 variant you find will be chambered in 7.62, or at least in a cartridge that uses the same size BASE as the 7.62mm/.308 (.243, 6.5 creedmore, etc..)
Almost any AR15 will be chambered in 5.56mm/.223, although there are some in 6.8mmSPC.
Personally, I would build a long range AR15 first, then maybe later an AR10. My biggest reason is parts availability, and cost. An AR10 will cost you about twice as much as it's little brother. And while you can build an AR10 yourself, I think it's actually easier to buy one then mod it yourself.
With an AR15 you can save a TON of money by building it yourself from the dirt up. You have a lot of choices too. I would go with a Rock River Arms lower reciever & a lower parts kit with a match trigger (any manufacture). The butt-stock depends on what you want, you could go with any of dozens of collapsable stocks on a carbine length buffer tube, or go with a rifle length butt & buffer system. The same goes for a pistol grip & fore-grip if you decide to mount one. There are literally HUNDREDS of options, and it all boils down to personal preference and of course LCF.
Their are probably just as many choices for your upper receiver as there are for a pistol grip. You could go with a "match" or "varminter" upper receiver, most manufacturers offer them. These come out of the box ready to be mounted & fired That reminds me; any AR15 upper & lower will fit together, regardless of manufacturer. Or, if you really have a boner for DIY jobs, you can order all the part & put them together, but the upper is best left to the pros. If DIY is the route you choose, the I would order a stripped upper & a complete bolt assembly (might as well go with RRA again) and a 20" barrel from Shilen with a chamber cut for 5.56mm, or Wylde. (Mystic player has Shilen barrels on the EE). You can order your quad-rail hand guard from a number of companies, since it's a long-range project, you're going to want to go with one that's free-floated. Check around, there are lots of companies that offer rails. The same with the gas-block, there are lots of different styles of gas blocks, some have rails, flip up sights, and other crazy gadgets. The installation of both the barrel AND a free float tube are best left to a gunsmith though, unless you're already familiar with the process. And of course checking the headspacing of the bolt to the barrel. There is also nothing stopping you from buying a complete upper reciever and then putting a match barrel & free float tube on it.
Finally, your add your choice of bipod & scope and you're good to go. Throw on some Back Up Iron Sights (BUIS) for LCF if you really want to.
I know that's all really generalized, but honestly, it has to be! When it come to putting together an AR15, there are a ridiculous number of choices out there. Regardless, I hope this helps a little.
Cheers.
Tim H