Can we see the stars at the edge of the Milkyway?
Take a look at the HR-diagram. You have forgotten to take into account that stars come in vastly different luminosities.  

For example, we can barely see all red dwarves out to a distance of 10pc, while we know that they vastly outnumber the more luminous stars.  

The luminous red giants however we'll be able to see on side of the edge of the disc, but even Gaia will not be able to look through to the other edge. It's limit magnitude is 20, while extinction through dust in the direction of the galactic center is often cited to be up to 100th magnitude.
Hubble deep field galaxies for comparison are around 30th mag.

However, there is Baade's window, a region just above the central galactic plane that has low dust content on our line-of-sight there. We'll be able to spy quite some stars there.

If you're interested in the Gaia mission, you can find information here.