Filling vias and holes on custom pcb
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.