Addressing only the physical layout of the board:
1. You have four mounting holes for a really small board. Two, one at each end, would provide all the support needed. That would allow you to reduce the width of the board by at least 1/20", probably 1/10" with some adjustments to the spacing of the components. You probably worked with a 1/10" grid but that grid can easily be divided by 2 or even 4 to get a 1/20" or a 1/40 inch grid. And, of course, you can go "off grid".
2. I see empty areas on both the front and back sides. The passive components (resistors, capacitors) are arranged on the front in a manner that limits the length of the board. You could squeeze their spacing a bit to get 1/20" or perhaps even 1/10" off the length. But the rear has a large amount of unused space so moving one of the components (resistors) from the front to the rear could easily get over 1/10" overall, perhaps even 2/10".
3. By using a 1/2 or 1/4 snap setting while using that 1/10" grid can allow you to space many of the components closer. Your PCB software should have a check built in to warn you when any spacing gets too small. Also, smaller trace widths can be employed in many of the traces in this circuit.
4. First, I agree that it is a lot better to use the "copper pour area" for Ground instead of +5. Using it for +5V only invites problems now and in the future. Along with #1 above, I will say that the +5 and Gnd pads on both ends of the PCB are spaced further apart than is actually needed. Bring them to a 0.1", spacing and if you are nervous about that, use oval pads like the ones for the IC. This, along with my suggestion of only one mounting hole on each end, will allow the +5V pad to be spaced further away from any ground traces and the mounting screws will be in far less danger of contacting it. Design it so that the Gnd pads are the ones that are adjacent to the mounting holes.