Dopamine rush? Heh, nerd. All the cool kids are doing serotonin.
Here's the deal with hordes, as unpoetic as I can make it:
Two things can be described as a horde: A faction at large or individual units in its armies. A faction that is in horde will get an announcement and will have a wheel(where did Buddhism come in?) under their insignia on their army banners. A unit that is horde will have a wheel in the upper left corner of its card, and will have "Horde" in its name.
A faction that is in horde has 0, 1, or 2 cities. In order to enter horde, you have to have 0. When you capture 1 city, you lose one half of your horde units, and every man lost this way will be added to the population of the city you captured. Happens again on the second city capture, and when you capture city 3 any remaining horde units will assimilate. Horde units cost no upkeep and do have uses (Horde Raiders have no non-horde substitute in Burgund and Lombard armies, for example).
In response to your actual questions - hordes don't replenish soldiers at all. Hordes can only lose soldiers. A horde can regain soldiers, in a sense, by settling for one turn, then abandoning the settlement, though I'm not actually sure that's legal, since I've never tried it. A horde disbands when you settle, it cannot recover. You can replace the horde soldiers with regular, standing soldiers after you settle, but while travelling, the only way to gain soldiers is mercenaries, which I don't count as recovering the horde.
Now, killing hordes is a tad tricky at times. But here's the thing - you don't defeat a faction as long as it has leaders. Assasins make the world go round.