Is it important that vias are plated on a PCB?
I Think you are using the term plating incorrectly. Plating will decrease the diameter of the hole, not increase it.

The larger dimension (the pad surrounding the via hole) is called the Annular Ring.
All the fabs I have worked with generally want a 0.005" per-side annular ring, or the via diameter + 0.010".

You definitely have some really hairy registration issues on the board you posted pics of. It may work, but you're really pushing it. 

Generally, you never want the via hole to break out through the edge of the annular ring, which is happening a few times in the picture you posted.

Anyways, Registration refers to the accuracy between a fab-house's etching and drilling process.
Basically, if a fab house etches a circle in the coper of a board, and then drills a hole in the middle of the circle, how close is the drilled hole to the center of the copper circle?

Remember, drilling a board and etching it are separate processes, and involve the board being unmounted and remounted in different equipment.
Generally, you can get as good registration as you are willing to pay for, and it looks like your boards are from a pretty budget board-house.

You need to allow enough annular ring that you never wind up with the via hole too close to the edge of the pad. This is generally specified by the fab house (they should have a minimum annular ring spec on their board requirements). However, it is important to remember that they may run your boards anyways, even if it does not meet their minimum required specs. 

The board house may run the boards anyways, and just refuse to fix any issues if they don't work out.
This is particularly common in China, where the general philosophy seems to be "Let the Buyer Beware".



Anyways, I think the reason you are finding your vias a bit odd looking is that you have tented your vias, which is the practice of covering the annular ring and the hole for the via with soldermask.

With 0.015" vias, you will occasionally get a contiguous layer of soldermask over the hole, and they will look different.