Electrical – CD4017 with multiple LEDs

4017 output is 'one hot', and won't source much over 1mA, so you can't use that directly.

If you must employ a 4017, then the simplest way to get to a base-5 chase sequence is to use diode logic to connect the 10 outputs to the bases of 5 NPN driver transistors. So output Q0 connects by 2 diodes to TR0 and TR1 bases. Output Q1 connects by 2 diodes to TR1 and TR2 bases, etc etc. Total bill of materials 20 diodes, 5 current limiting resistors on the bases, and five transistors. If you use a driver array like ULN2803, then you don't even need the base current limiting resistors. That array goes up to 50v, so you can put LEDs in series to reduce the current requirement.

Finally, use an oscillator to drive the clock pin. A variable speed clock varies the speed of the chase.

The 4017 is clocked on the rising edge, and you have the clock line high when it comes out of reset.

Try connecting clock in to Vdd and the 555 output to inhibit in. Inhibit is just an inverted clock input (sans Schmitt trigger). Or add an inverter between 555 and clock in.

In general this is a really crummy reset circuit - criminally bad for anything important. Use a reset (supervisor) chip that provides a sharp fixed-length reset pulse out (eg. 200ms) and also detects slow brownouts and slow Vdd rise. They are plentiful and cheap, and designing one that is bulletproof is non-trivial.

If you insist on using this circuit, at least add a few K resistor in series with the reset input. Otherwise shorting or putting a heavy load turning Vdd off could damage the chip by discharging the 4.7uF through the input protection diodes.

Edit: Rough schematic showing supervisor chip ADM803/ADX803, you may want to add a power-on LED or resistor from Vdd to GND to help discharge the 5V faster so it resets reliably on a short power interruption.