1. You can be reasonably sure. Try setting your router to open auth and see if you can fake auth, then try ARP injection and see if the increase in #/s correlates to the number of packets you've sent.
2. Airodump is passive. You're simply capturing packets sent by associated clients.
3. Check tcpdump. If you see a bunch of deauth packets then you aren't gathering IVs so it's just noise.
For packet injection to work, you're going to have to turn off your monitoring interface and set the MAC to the same as one used by an associated client.