Will cleaning extra well (flossing, brushing, tongue scrape, antiseptic mouthrinse) help the inflammation heal faster at all?

Most likely: If we are talking about gum inflammation then yes. Gum inflammation or gingivitis is usually caused by the continued presence of plaque biofilm and tartar along and/or below the gumline. Once these irritants are remove (by a dental hygienist) then the inflammation goes away. After that keeping your teeth clean with good oral hygiene keeps inflammation of the gums to a minimum.

Yes: I am assuming you referring to gum inflammation. If that's th e case, then yes. Good home care will cure early inflammation. If the condition persists, see your dentist or periodontist.