Electronic – ATtiny85 to control a relay, which will turn on and off a motor

I am trying trying to control a 5V relay (SRD-05VDC-SL-C) with the ATtiny85. I have a simple sketch uploaded (Blink without Delay example):

const int ledPin =  0;      // the number of the LED pin
int ledState = LOW;             // ledState used to set the LED
unsigned long previousMillis = 0;        // will store last time LED was updated
const long interval = 1000;           // interval at which to blink (milliseconds)

void setup() {
  pinMode(ledPin, OUTPUT);      
}

void loop()
{
  unsigned long currentMillis = millis();

  if(currentMillis - previousMillis >= interval) {
    previousMillis = currentMillis;   

    if (ledState == LOW)
      ledState = HIGH;
    else
      ledState = LOW;

    digitalWrite(ledPin, ledState);
  }
}

The whole thing is being powered by an 18650 Li-Ion battery (3.7V-4.2V) and the measured input is averaging 3.88V. Here is a quick sketch of how the circuit looks like:

Some issues comes up WITHOUT the motor connected:

1. If I connect the battery and relay directly, you can hear it click, despite it being rated at 5V input.
2. If I try to trigger the relay from Pin0, the oscilloscope will show the voltage of Pin0 to be about 2.72V, when the relay looks to be switching. If I replaced the relay with an LED, the voltage of Pin0 is about 3.88V. What causes the drop in voltage? I believe the current should be enough since it's an Li-Ion battery.
3. What sort of protection do I need to implement in this circuit?
4. Will the Li-Ion battery be enough to power this circuit? If I connect just battery, relay, and motor, it runs fine. The issue comes with the addition of the ATtiny85.