A couple of ideas :
You have put in a lot of work into this, have you used a PCB design prog ?. 30 thou is a bit small, I use tracks as big as I can. Trouble is if the power feeds all the LED’s first, then goes to the logic chip, any volt drop due to the LED’s switching will affect the chip's power rails, and in a bad case can cause it to randomly reset. In fact I always use a ground fill, this fills all unused areas with copper at 0V so keeps everything close to ground which keeps it stable. Also less copper to etch away so less pollution.
Looking at the bottom layout : You are going to have to use a thin track between the central IC pins 10,11 (?), nice to get it outside the IC, means it can be thicker and less prone to damage. I like to keep traces as thick as possible to avoid chance of a pad or track being lifted due to heat when soldering. You have two red tracks next to the RHS of D19, they are thin and quite close together. If you move the top one up and pull the bend on the lower one down you can double track widths. Layout, as a first step why not move the output connector to the edge of the board it will be easier to get to.
Looking at the circuit, all the diodes seem to be purely signal routing, feeding the signals from the counter to the transistors, so 1N4001's is definate overkill. You could use basically any diode here, 1N4148 or similar would work fine. In fact........
Has anyone figured out what diodes D12, D13, D14, D15, D16, D17, D18 and D19 are doing ?. If its an attempt to reset the counter when it fills up, shouldn't a wire go to the reset pin ?.
Also on the circuit diagram D4 and D8 cathodes connect together but don't go anywhere. Should they go to R4 instead of R4 going to R3/D3/D7 ?
I think getting good at PCB layout comes with practise, and having to build what you design. I had several goes using a Daylo pen and baths of ferric chloride and over etching boards and having to patch them up with jumper leads !.
Ken