I'll try to answer each of your questions separately.
- No, you don't need multiple LDO's. You could get away with using 3.3V for the entire circuit as long as your MCU is 3.3V compatible and not 5V only. In the datasheet where you read the voltage could also be 3V, that's a different relay from the same family. The 5V relay has a maximum actuating voltage of 3.75V, this means that it is possible that 3.3V will actuate the relay, but sourcing a 3V relay from the same family would be a much more reliable alternative (HK4100F-DC3V) as you can't guarantee the 5V relays will work every time, you might order some which work fine and then order the same again but they don't work.
- Yes, the current that will be drawn by the relay will be the supply voltage divided by the coil resistance
For your 2.2V LEDs, remember that this is the forward voltage of the LED. Try not to think of it as an operating voltage. The LEDs will drop 2.2V across them but will draw as much current as they feel like until you end up with a dead LED. You need to put a resistor in series with the LED that will drop the remaining 1.1V (If we use 3.3V). We then select an appropriate resistance as to limit the current flowing through the LED, this can be done using the equation \$R = (V_{cc}-V_{f})/I\$. Where Vcc is your 3.3V Vf is your 2.2V LED forward voltage and I is your required current which you can find from the datasheet.
The other thing to keep in mind is that dropping from 9V down to 3.3V at the current you require is going to be about 1W power dissipation, if you do use an LDO you'll need a heatsink for that more than likely. The other alternative would be to use a switching regulator instead (TSR 1-2433 is the sort of thing I'm talking about)
Looking at the current consumption of this device, if you have the 100mA drawn by the sensor, 20mA from the LEDs and 20mA from the relay coil then your 9V battery will need to source ~150mA. 9V batteries are around 550mAh, this means your circuit is, at a worst case scenario only going to run for about 4 hours