Electronic – One particular part number not showing up on Eagle board layout

1. You're using a strange mix of curved, straight and 45° traces. The point is purely aesthetic for a design like this, but I would try to stick to 45° traces.
2. Yes, they're the solder pads.
3. Not if your PCB netlist matches the schematic's.
4. You'll have to choose actual parts first and check the drawings in the datasheet, or, if you have parts already, measure them. This goes especially for the capacitors, resistors are more forgiving about this (though you'll want to use as much as possible the same pitch for them).
5. If you don't want to place a header or connector there I would place them a bit more apart. You can use a 1-pin header as component.

Also try to be consistent with the symbols in your schematic. You mix American resistor symbols with a European potmeter symbol. (Personally, I don't like the American symbol at all; a schematic with lots of resistors IMO looks messy and frenetic, while the same with European symbols radiates rest & relaxation :-)).

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A comment on the double-sided board. This is a design that easily can be done on a single layer. It's worth practicing the layout part of your design. You'll learn how just swapping two components may simplify your routing significantly. For instance the red trace from R6 to the IC can be swapped to the other side by simply placing R6 between R4 and C4. Things like that. It will help you later on when you have more complex designs with hundreds of nets. And it can be fun, too. I like this kind of puzzles a lot. It's like playing planarity.
The problem with this double-sided design is that there are pads which are not accessible for soldering, because they're under a component on the component side, like with C3 or M1. If you don't have plated-through holes you can't be sure the wire is properly soldered to the pad. Worse, if the component's package sits on the pad you can almost be certain that there won't be a solder connection between pin and pad.

Create a footprint with GND and AGND pads. Draw copper between these pads. Yes, this will produce a DRC "Overlap" error as shown below:

This is OK. There three buttons at the bottom:

• Clear all
• Processed
• Approve

"Clear all" will temporarily clear the list for this run of the DRC. I'm not sure why that's useful; just close the window if you want it shortened.

"Processed" will fade out the color of the red X. This is potentially useful if you're iterating through a long list of DRC errors and fixing them as you go; you can keep track of the ones you think you've corrected.

"Approve" is the only one I use on a regular basis. This moves the error from the errors list to the approved list:

and keeps it there on subsequent runs of the DRC. Note that this only moves this specific error with this specific pair of nets at this specific location. Closing this window and running the DRC again produces the notification "DRC: 1 approved errors"

and no "DRC Errors" dialog. You can get this dialog back by creating an error, or (preferably) the errors command, the yellow exclamation point in the above screenshot, or the menu Tools -> Errors.

The "Approve" functionality exists for a reason, the same reason that we have tools like

#pragma GCC diagnostic ignored "-Warning"

Sometimes, it's OK to ignore a DRC error. This is one of those times.