Each Maneuver Must Be Evaluated Individually...
There's no way to generalize the applicability of every maneuver to every ability that grants additional attacks. Some can be used and some can't.
In your example, while the 1st-level Desert Wind martial maneuver burning blade (boost) (ToB 52) does last, like its duration says, until "End of turn," but the 2nd-level Stone Dragon martial maneuver mountain hammer (strike) (ToB 84) says, "As part of this maneuver, you make a single melee attack." That really means only one single attack gets the benefits of the martial maneuver mountain hammer, even if the target's dropped by a creature using the martial maneuver mountain hammer who has the feat Cleave (PH 92). That's because buried in the Blade Magic chapter in the section (Type) under the heading Strike there's this text:
Because strikes allow for a specific form of attack, you cannot benefit from spells or effects that grant you extra attacks when making a strike (such as the haste spell or a speed weapon). (43)
Thus an adept possessing the feat Cleave who initiates the martial maneuver mountain hammer who drops a foe does get to make the attack granted via the feat Cleave, but he gains no benefit from the martial maneuver mountain hammer when he does. To be clear, when he's granted that extra attack from the feat Cleave, the adept isn't benefiting from "extra attacks when making a strike" but benefiting from extra attacks after resolving the strike--that is, as a result of the strike.
...But Many Boosts Last Until the Turn Ends
In the heading labeled (Type) the section Boost reads, in part,
A boost is a maneuver that grants a bonus, often on attack rolls or damage rolls, for the duration of your turn. A boost always requires a swift action, usually allowing you to initiate it before unleashing a standard action or a full attack. Some boosts impart additional effects, such as stun or fatigue, to your attacks, and others provide some additional effect on an enemy you have just successfully struck in battle. If a boost affects your attacks, it applies to all of your attacks for the round in which it was initiated, but its effect ends at the end of your turn. A boost’s effect applies for its duration, no matter which weapon you might wield in that round. Even if you switch weapons in the middle of your turn, the effect of the boost applies to your new weapon as readily as the previous weapon. Each maneuver’s description gives you the details of each boost’s effect. (ToB 42)
Emphasis mine. Therefore the martial maneuver burning blade lasts until the end of the initiator's turn because it affects attacks, but the duration of 6th-level Diamond Mind martial maneuver moment of alacrity (boost) (ToB 64) remains instantaneous.
They are special nonmagical techniques based on being a scholar of combat. This is explained in the Battle Master entry:
Those who emulate the archetypal Battle Master employ martial techniques passed down through the generation. To a Battle Master, combat is an academic field...
Individual maneuvers explain what you're actually doing:
Commander's Strike: You can forgo one of your attacks... to direct one of your companions to strike.
Goading Attack: You can... attempt to goad the target into attacking you.
In each case, you're mundanely directing your ally or goading (through voice or gesture or just annoying attack placement) your enemy. You're just really good at it because of your battle mastery.
Although I can't find a reference in the PHB, it's generally accepted in D&D that you're doing more with your turn than your single attack. Your PC might constantly be feinting, dodging, making threatening jabs, and so on. Your attack rolls represent those actions that are actual opportunities to do damage. From this perspective, then, a Commander's Strike doesn't mysteriously make your ally move faster; it just gives them an extra opportunity that they otherwise might not have had, just like a retreating enemy lets them make an opportunity attack.
In the games I've played, these maneuvers have been explained however seemed appropriate in the situation and were never assigned advantage or disadvantage, but I don't see a problem with the saves being modified in unusual circumstances. I'd just make sure that you're treating other players' unusual abilities with the same level of scrutiny.
P.S.: The Battle Master is, in part, 5e's version of the Warlord class from fourth edition. That class has the Martial power source, indicating that it's wholly nonmagical, with the following explanation:
You have become an expert in tactics through endless hours of training and practice, personal determination, and your own sheer physical toughness.
Best Answer
Quoth ToB, p38:
P39:
...
P42:
So an example of a normal strike is: "blistering flourish" from desert wind. If you hit, you glow for a minute and creatures nearby need to make a DC 11+wis or be dazzled.
death mark that explodes in a fireball is the same.
Looking at diamond mind, the action before thought counter requires a concentration check in place of a reflex save, no extra attack.
Diamond nightmare blade is the item you described. Unfortunately it looks like case one from the text and as discussed here, you roll concentration to get a 4x multiplier and make your opponent flatfooted. You then have to attack as normal.
Functionally this is "roll twice to do OMGWTF damage", which indicates that you should invest both in a very high concentration (or other key skill of the discipline) and a good attack bonus. Given that insightful strike replaces your damage with a concentration check... and the insane ways of boosting skill checks... this doesn't seem like an unreasonable requirement.
Every power describes its use. Most do not "replace" attacks save where they indicate that they do.