DIY LED stand wiring
You're treating the LEDs as if they're (2.5 V - 3 V) lightbulbs. Lightbulbs do not mind if the voltage is a bit lower or higher but LEDs do.
LEDs respond very strongly to the applied voltage. Apply a voltage that is too low (5 V for 2 LEDs in series) then almost no current flows and the LEDs burn very dimly. Apply too much voltage ( 5 V for 1 LED or all LEDs in parallel) then way too much current will flow and the LEDs will overheat and burn out.
The fact that your 5 V supply is rated for 500 mA doesn't matter as the LEDs will want to draw much more than 500 mA and the supply will be overloaded. If you're lucky the supply will switch off. If you're unlucky the supply could catch fire.
You should NEVER apply a voltage directly to a LED. Instead, they need a current. what is also OK is to connect a series resistor to limit the current and then apply a voltage to LED + series resistor. The resistor will limit the current and prevent the LED from destroying itself. How to calculate what resistor is needed (value and power dissipation) can be found on this site as it has been asked many times before.
Example of a quick calculation: if one LEDs needs 3.5 V, that means there needs to be 5 V - 3.5 V = 1.5 V across the series resistor. You want 20 mA current per LED that means a series resistor of 1.5 V / 0.02A = 75 ohms in series with each LED. That 75 ohm resistor will dissipate 1.5 V * 0.02A = 0.03 W which is quite low so any resistor can do the job (without getting too hot) even a small 1/8 Watt resistor.