arduino resets or hangs after providing power to DHT22
The sensor is a large capacitive load. You need to improve decoupling of the CPU power supply.

A 10 uF between 5V and GND will help for the current spike and the resistor in series with the sensor will not be an issue (50 uA on 250 Ohm makes only 12.5 mV voltage drop once the sensor is started).