Good boy with a longer nap today! That should help things a bit. His bad nights and those 45 min naps have probably made him OT, which makes the nights even worse than teething alone.
I would say that based on your routine it might be time to stretch those A times out a bit, especially the first one. He is probably short napping because he has only been awake for 3 hrs so goes down UT. At 9 months he "should" (of course, each babe is unique) be able to do about 3.5 hrs A time. If you tweak his routine, I really think that it will even out his nights. It might not fix things right away because teething is such a factor, but it should help. Poor monkey, cutting 4 at once! Ouch.
Don't feel bad if you have to feed him; don't feel bad about using APOP to get through this. You will get back on track. I have BTDT with Nathan, and I know how hard it is to try so hard to settle your LO at night when they just will.not.stop. crying! There is zero point in trying to start sleep training when they are ill or teething, so just tweak his routine a bit and do what you have to do until he feels better. Then you can fix any bad habits that have sprung up.