Abscesses can be due to accidental trauma, such as a bash to the wall or stone bruise to a weak sole - like bashing your nail with a hammer & getting a blood blister which can become infected. They can be due to constant compression - eg. shoes create 'corns' which can abscess, contracted heels can become abscessed especially in the bar region, etc. Or they can be due to other trauma/infection such as 'seedy toe', laminitis, leverage on walls, etc. They can also sometimes be due to puncture wounds.
If it were a 'one off' type affair, they usually resolve pretty quick. But if it's a 'symptom' of other problems, it can perpetuate. I suspect that as your horse also had an abscess burst(may or may not be literally connected), it's likely that something else is going on. Some hoof pics & more info might help us give you a better idea & more specific answers.
As a rule, I disagree with digging for abscesses, and I wouldn't tend to 'plug holes', as this can make for a nicer germ growing environment if you haven't got rid of every bit of infection. There are exceptions though, such as if there's a hole in the sole that may need protecting.
Bluebird a bit of cotton wool is not likely to cause any building of pressure & assuming there wasn't separation/infection all the way up anyway, it's not going to 'track' through healthy wall/laminae all the way to the coronary. While it can be a 'better of evils' with regard to need of relieving pain, and some vets are quite free about giving bute, an anti-inflammatory is not the best first aid IME, as the inflammation is, as you put it, the body's defense mechanism, so suppressing that can suppress the infection & cause it to perpetuate and potentially cause other issues.