Calculation of Relay current
The BJT can't pull its collector below ground.

Instead it will go into saturation mode with \$V_{ce}\$ in the neighborhood of 0.2 V. So the voltage across the relay coil will be around 4.8 V. And the coil current will be about 40 mA.

If you buy a 5 V relay, this will be exactly what it expects and it will work accordingly.