What does EMSE mean

Simply put, toe and heel are the areas of the solder joint that extend from the front and rear of the lead/pad. Think of a solder pad on your PCB as a piece of paper, and think of your hand as one of the leads coming off an IC package. If you place your hand in the middle of the piece of paper, the "toe" and the "heel" are the distance between the tip of your fingers, and the palm of your hand, to the edges of the paper. That's the extra solder in the joint that extends farther than the actual lead/pad on the IC itself.

Here, they are recommending a specific amount of toe because it will provide mechanical reinforcement for the joint. Since there is no room for a suitable heel, given that this is a castellated package that will sit pretty much flush against the main PCB, the only way to add more solder to the joint is to expand outwards, and thus they recommend to give it some extra toe.

To comply with their recommendations, you'll have to turn off the default stop layer and cream layer for the pads, and draw them by hand. This is as easy as going to the properties for the pads to turn both of those features off, and using the rectangle tool, selected to the tStop and tCream layer, to draw the features.

Based of the picture, it sounds like they recommended a tStop that matches the recommended pad exactly, and a tCream layer that matches the pad exactly except for the edge opposite the package, which would extend out 4 mils further.

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.