Let's start with the last question first. At Elcrow, you won't save anything by building a single layer board compared to a 2-layer board. If you do make a single layer board, I don't know if they plate the holes by default.
This is similar to the board you showed originally with non-plated holes. The one advantage of non-plated holes are that it's easier to unsolder leads from the board because there's no solder connection inside the holes. The solder is only on top of the copper. This type of board
could be cheaper to have made, but this is not the case at Elecrow.
The board below has plated holes, and incidentally is 48 cents at Elecrow. The advantages are it's easier to solder since solder flows into the holes and makes a solid connection and you can make easy connections by soldering a fine wire between pads on either the top or bottom.
My preferred style would be a layout like this board, with groups of holes connected together with traces. A 16mil (mil = thousandth of inch, not millimeter) or 24mil trace between grouped pads is plenty wide enough. The green color on the above board is soldermask. It pretty much repels solder, so inadvertantly bridging pads is more difficult.
The pads on the above board are also solder-plated, which is what you'll get from Elecrow; it's better than the bare copper of the first board shown because it resists tarnishing, which makes soldering difficult.
As far as pad size goes...it depends! Standard square header pins fit in a 0.04" drilled hole, allowing for plating thickness. That's probably a safe size for most components on a protoboard but check the manufacture's recommended "pcb footprint" (or look in Eagle libraries) for parts you might use to see what's recommended.
As to which is
better, it depends on what you want. I pretty much make a mess of protoboards when I use them!