Electronic – this structure on PCB

EAGLE has no say over the outcome of the thickness of the actual copper layers themselves, since you'll be etching the board yourself and you're the one picking the copper board stock. However, you can increase the size of the annular ring to give yourself more room to drill a hole close to the center of a PTH.

Using the Info tool (the button that is the letter i), click on the PTH you are interested in and adjust the Diameter from auto to another value that is greater than your Drill size. Click Apply and visually inspect the hole - if it's still not as much copper as you want, increase the Diameter again until it's as thick as you want.

If you want to do this to all your PTHs, just use the Change tool, go to Diameter, and click the ... option at thebottom to set a custom value. This would ideally be the Diameter you chose when playing around with a single PTH. Then just start clicking on the other PTHs you want to update them.

This will only work, however, on PTHs you defined manually on the board layout. If some PTHs are part of the part's package, for example, you will have to edit their Diameter manually in the package itself, not in the board layout.

Could be leaky, could be pickup of AC hum from the mains or RF from AM radio etc. as the transistor will act as a detector.

To tell the difference, put a 1uF ceramic (low leakage) capacitor from base to emitter and see if the issue disappears.

In any case, you can reduce the sensitivity simply by connecting a relatively high-value resistor from emitter to base. Something in the range 10K-100K will likely work for you.

Edit: One thing that can cause a great deal of leakage is the use of acid-core (plumber's) solder. If this is accidentally used, clean the board with warm soap and water, preferably followed by isopropanol alcohol. I have seen mains circuits burn up as a result of this. As an aside, the solder used in plumbing was often 50:50 PbSn, so it would not solder as nicely (and will require much higher temperatures) compared to good electronic grade 63:37 solder.