This must be so frustrating for you. You are right to say that you can't just carry him around all day. That seems to work for some, but really it is pretty un-doable for most. My back just couldn't take it! Here are a couple of thoughts that may help:
Try not to worry about what is, or is not, getting done around the house. The stress you feel about this is conveyed to your boys and can really make things worse. Try to fully switch your motivation to get your son to sleep to being all about what is best for him. He needs adequate sleep for his health and development, not because you need to clean the house. :o) I know it sounds like a small thing, but the way we think has a big read-out for our lives and in those around us. If your hubby is part of this stress, just let him know that the house will be a bit disordered for some time, but you will get a handle on it after you get your little guy sleeping better. If he is really bothered by it, he can pitch in a little more for a while. :o)
Next, really take a good look at what is happening. If he is falling asleep while eating then waking after a short time, he may be actually hungry again. With my girls, I did everything humanly possible to keep them awake for a full feeding. If they started to doze off, I'd take away the breast and pick them up, unwrap their blanket, whatever it took to keep them awake. A good full tummy is key to a nice long nap! You may be in a 'snacking' cycle instead of a 'meal' cycle. Also, he is certainly old enough to start solids (if you haven't already). Rice cereal, etc. may help him to stay full longer.
You don't say how he is sleeping at night. At 6 months, he should be sleeping through at least 9 hours, or up to 12 hours. If this is happening that is great. At least you get some sleep. :o) He should also be taking 2 long naps a day. About 1.5 to 2 hours each.
Another thing to really look at is how/where he is sleeping. Assuming he has a quiet, fairly dark, place to nap, try letting him fuss a while before you decided he is really awake. Babies (and adults for that matter) have times of deep sleep and times of lighter sleep. In adults this lighter sleep may be marked by tossing and turning a bit, but in babies it can be marked by fussing/crying. If you just leave him he should settle back down and sleep some more. Try letting him fuss for at least 10 mins before going in to get him. It will be a long 10 mins, but totally worth it. My youngest sometimes took almost 20 mins to fall back asleep, but she always did and slept another 45mins - 1 hour. Try this and see if it works for him.
Re-training him to get a good amount of sleep will take some time, so don't give up. Decide what your plan is and stay consistant with it. He'll catch on.
As for the teething, I used Hylands teething tabs most of the time, but baby Motrin or Tylenol when it really got bad.
Good luck with your parenting adventure!