If someone is affected by this spell and the aura of conquest they would indeed have a movement speed of zero and waste their action. It's an extremely powerful combination that I'm going to use as much as possible once my Oath of Conquest paladin reaches level 9.

The spell for letting the level 7 Monk know that "No, you are not, in fact, safe from being frightened"

Only to a degree. The frightened monk could use Step of the Wind to take the Dash action and then Stillness of Mind to end the fear spell. This was confirmed by Crawford 7 years ago (https://twitter.com/JeremyECrawford/status/604496205435895808). However, he is not the boss of me and since this is a specific class feature that is part of what makes a monk a monk, I have ruled that the monk's Stillness of Mind Action is used to override their desire to flee as an Action. Play it as you see fit, but this is one of the things that defines a monk and should not be taken away by rules fiddling IMHO.

That still requires a ki point. It was more to the point that you're not immune to the condition in the same way that you are immune to poison at level 10. 
I'd also disagree that a specific spell "takes away" anything. Spells define casters, yet counterspell and anti-magic field exist. All abilities have counters, and not all sources of charm and fear force you to use your action. Its also hardly the most debilitating spell for somebody that 1. Is fine with being disarmed and 2. Likely already does quite well on Wisdom Saves.

Ultimately I don't have any major qualms with using your bonus action to satify the requirement, but I do think that if you go that way you also need to enforce that the Dash has to happen BEFORE any other actions to avoid ending up in a scenario where you have characters promising dashes they may or may not end up having and ending up in "Dash Debt"

What is the DC? What would a player need to roll on a saving throw for it to be considered a successful save?

I believe 9th level Gloomstalkers can also use this according to Wiki dnd 5e 

http://dnd5e.*******.com/ranger:gloom-stalker

I think this spell needs better specifications. Because if you have the sentinel feat and they dash away which they have to, their speed becomes zero when you hit. For oath of conquest, their speed is zero as well. Are you telling me if they fail and you have sentinel feat or oath of conquest you won the fight basically? There are also other ways to make their speed zero by someone makin them prone constantly. This spell can be very broken whoever interprets it. 

I'd argue that the creature would still have to take the dash action as, while the creatures speed is zero, there is still somewhere to move, the creature just can't move to it. Also the spell is one of the oath of conquest paladins' oath spells, so the designers likely intended for them to be used together