Where to dispose of infectious waste from a private home?

Regular trash is fine. If you are worried about any other possibly infectious material, flush it down the toilet. This is basic sanitation of all waste.

Think about it. Meat goes in the trash when it is spoiled. The bacteria that grows on it then is one of the strains that could have abused your abscess. This is why digging through trash cans isn't advisable.

There is very little risk from your wound dressing. It's about the same as throwing out a bandaid.

It would e a little different if it was a needle or something else that could easily break someone's skin and carry something inside that cut or puncture.

It's ok to just throw it in the trash. If there were any special procedures that had to be done with it, you would have been informed of this before being sent home from the surgery. In any case, it's no different from disposing of a bandage or something similar; it doesn't need any special procedures unless you were infected with something really nasty.

If you have a fireplace, burn it if you want to, but it isn't necessary.

I suggest burning it in an old coffee can. Obviously, you want to be sure that it is done outdoors, and is well away from anything flammable. Burning kills any pathogens, and all you have left is a bit of ash. It's pretty much the same thing that hospitals do with medical waste, just on a smaller scale.