Multiple converters off of one power source is a pretty common thing. A modern server motherboard gets its primary power from a 12VDC power supply and has to convert that to a number of voltages including +5V, +3.3V, +2.5V, +1.8V, +1.05V and +1.0V. (Sometimes there may even be more voltages or there could be duplicates of one or more of the above to deal with multiple CPUs or multiple DIMM memory banks. In a laptop a similar selection of voltages have to be produced from the laptop battery pack.
Yes, this is certainly possible and very common. I'd make sure there are correct capacitors before each of the boost converters just to make sure they don't start doing anything weird (then again, I'm kind of a throw in tons of capacitors just in case type of guy). If you buy a boost converter board it will typically have these built in.
The other thing to keep in mind is that you might not want to do this. For instance, if the 5V device is something small that will draw very little current, you could just use a single boost converter to make the 3.7 into 12, and then use a linear voltage regulator to bring that down to 5V. It's not the most efficient way but it is easier and cheaper and sometimes smaller; and if it will only be drawing a few mA, then it might be a small enough inefficiency to not worry about. Depending on the specific boost circuits used and current drawn, it might be better to boost from 3.7V to 5V and then boost from 5V to 12V. You'll need to look at efficiencies in the data sheets for the specific current you need.
Also make sure that the boost converter can handle the current necessary and remember that if you need 1A output at 12V, then you'd need approximately 3.5A input at 3.7V (depending on efficiency).