The extension cables are probably best for you. Basically the 1x slots are designed for cards besides GPUs, like NICs and soundcards.
For gaming, these 1x slots would not work with a GPU. For mining, however, we do not need the massive bandwidth that a 16x slot provides, so you can use the card with only the 1x section of the card.
Since the graphics card has the full pins for a 16x slot, you would have to file off the right-side of the 1x slot so the card will physically fit into the slot.
You can either get a 1x-->1x extension cable, then file the end off of that, or you can get a 1x-->16x extension cable that you won't need to file.
The only way to get a bunch of 16x slots on one board is to get a higher end board like the MSI one everyone here uses. If you do this, you'll have heat problems anyway, so probably best to start with 2x cards in 16x slots then add more via extension cables away from the other cards if you want to expand.
Awesome! Thanks for the explanation. My only question is, if I get a 1x-->16x cable extension, where do I put the actual video card? I guess that would depend on my case?