Electronic – What’s this special type of PCB prototyping board called

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.

Congratulations. Welcome to the club.

You MUST have through wires in vias.

• Solder "glops" will not work. If they work sometimes they will not work always, and one which do happen to work will die during rework/touchup/alterations.

• Threading some single strand wire through the holes and then soldering it and then cutting it off is advised. Do not use multi-strand wire - tends to make larger ends, harder to deal with overall - no better result.

• Wire can be left vertical in hole and soldered and clipped as above. this "tends" to stay in place when resoldered for many values of tend, but an enthusiastic hot irom may draw it out. For extra safety bend wire at right angle on each side before soldering. Clipping before soldering leaves a smaller solder blob than otherwise.

• You can buy special pins made for this purpose but they tend to be ugly, cost money and work no better.

• You can use dressmaking pins for this but make sure they they solder OK. Nickel plated brass ones solder well (or, the one that I used long ago did). Solid steel pins are often very very very hard to solder). Small head sizes are desirable - large head sizes take up more room than needed.

• I've seen people suggest running a wire through the same hole as an IC pin and using that to gain access to the connection on the top side. I've not tried that and it sounds messy but I mention it as a possibility.

With tracks that go to the top of IC leads you must either

• Solder to track on top of PCB or

• Provide an alternative link from top to bottom as well (via a "via" or other.)

Some devices are easy to top solder to, some aren't. If wires come to IC pads from under the IC, having enough pad on the outside to solder to is wise (at least).

If using sockets (and they are often a good idea as long as not rubbish) then choose a type that allows iron top access to the top of the pin when inserted. "Machine screw" round pin sockets are generally well spoken of and work well enough for this purpose. I've had good results from them over the years.

Whatever socket you use, choose an acceptable quality one. this need not be the dearest but will probably also not be the cheapest. Here brand name often counts. Cheap and no-name is often no-quality.