- Is a good idea to run the Arduino on 3.3V? (I thought so since my wifi module also runs on 3.3V)
Won't hurt anything.
- When running the Arduino on 3.3V, can I run it on 16Mhz? (I hope so, otherwise I'll need another crystal and change the bootloader)
Technically, no. Practically, probably.
3.3V / 16MHz is outside the processor's specifications. However, many people run the processor at that point without problems. There are Arduino compatible boards available that run at that point.
If you're building something that has to be guaranteed reliable, pick a different point (higher voltage or lower clock frequency). Otherwise you'll be fine.
Someone else will have to help with the rest.