Yes, but I noticed something else.Look better?
Correct me if I'm wrong, but won't the outputs of the 555 timer and CD4017BE be close to supply voltage (3V from the cr2032)? The LEDs I have picked only need a forward voltage of 2V (for the LED on its own) and 2.2V for the sequential LEDs.Yes, but I noticed something else.
If you using only one CR2032 battery it will barely light an LED.

Depends upon how long you want it to run before battery replacement and the value of LED current.What would you suggest instead?
If I increase R1 and R3 to 350k and 10k respectively, that would mean the time high would increase to around 2.5 seconds. Are you also suggesting I decrease the capacitor to 1uF?You can increase both resistors by 10x or more to get this current way down.

- That is a potential, but I'm worried about space + price.In addition to using a CMOS timer, you should consider using a 9V battery and low current (2mA), high efficiency LED's.
Then the LEDs can be driven directly by the CD4017B. A 3V supply is very limiting for this application.
You haven't written how fast the LEDs should flash.
It's a potential, but like I said, I'm worried about space constraints. What would the cons of 3V (1 coin cell) or 6V (2 coin cells in series) be?The board is large enough that you could have a 9 V battery on the back side (opposite the side with the LEDs).
70.247 * 149.740mmWhat are the outside dimensions of the board?
That is the result of using low voltages....components get more expensive, because they need to be more efficient, with less losses.- That is a potential, but I'm worried about space + price.
Not necessarily....there are high intensity, low current LEDs.- I can certainly choose lower current LEDs, but that means lower brightness, from what I've seen when looking.
250ms? is that between flashes?- The timing for the 555 should be ~250ms, or as close to it as possible.