Horde Breaker reads (emphasis added)...
Once on each of your turns when you make a weapon attack, you can make another attack with the same weapon against a different creature that is within 5 feet of the original target and within range of your weapon.
...so it can only be used once after taking some other attack. There's no "repeat" and it's not separate from the initial attack in any way. It works much like the Extra Attack ability, just with special conditions. No matter what triggers it, it is once on each of your turns and only grants one attack.
Also note there's no such thing as a "full attack round" nor does an attack have a "bonus round". There are actions and bonus actions. Try not to think of "bonus action" as "an action that's a bonus", try to think of it like "bonus action" is a single word, like it's a separate thing in and of itself.
Martial Arts grants an additional attack as a bonus action.
When you use the Attack action with an unarmed strike or a monk weapon on your turn, you can make one unarmed strike as a bonus action.
Flurry of Blows grants two additional attacks as a bonus action for the cost of 1 ki.
Immediately after you take the Attack action on your turn, you can spend 1 ki point to make two unarmed strikes as a bonus action.
Monks get Extra Attack at L5:
Beginning at 5th level, you can attack twice. instead of once, whenever you take the Attack action on your turn
The character could make your normal attack (two via Extra Attack at L5) and one attack via Horde Breaker (assuming a valid target) as an action. It could then add one unarmed attack via Martial Arts (or two with Ki points via Flurry of Blows) as a bonus action.
Maximum possible attacks, assuming all conditions and resources are available is five. Nothing multiplies or triggers more than once. Nothing chains together more than described above.
SCAG Weapon Cantrips
Interestingly, Horde Breaker does not specify that it is triggered by the Attack action, just by a weapon attack. A character with access to some of the SCAG weapon cantrips could trigger it off that because they involve a weapon attack. The Horde Breaker attack wouldn't benefit from the cantrip's magic, but any weapon attack is enough to trigger it, and those cantrips include one.
Horde Breaker has the same "once per turn" (rather than "once per round") wording that Sneak Attack has. One could use the same Haste/Ready action technique: use the Haste-granted attack on your turn, adding Horde Breaker, then ready an action for somebody else's turn, taking your normal attack and Horde Breaker again when the readied action is triggered.
Movement (or "Who Gets Hit")
As far as attacks go, assuming you meet the requirements to make the attacks, you're allowed to freely move between attacks. 5E puts no restrictions on that.