Quoth ToB, p38:
Initiating stances and manuvers:
... The process of initiating a maneuver is similar to that of casting a spell.. though there are some key differences ...
P39:
Resolving a maneuver or stance:
Many maneuvers include an attack of some kind. All offensive combat actions... all maneuvers that opponents can resist with saving throws or that otherwise harm or hamper subjects can be considered attacks.
...
P42:
Strike: A strike is a maneuver that allows a special attack. Strikes almost always require a standard or full-round action to complete. Many of them involve making a melee attack as part of completing the maneuver. If the attack hits, your opponent normally takes normal melee damage, as well as suffering the effect of the strike. When making a strike, you use your BAB, [and normal modifiers] as normal.
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.
You must make a Concentration check as part of this maneuver. The DC of this check is the target creature’s AC. You then make a single melee attack against your target, also as part of the maneuver. If your Concentration check succeeds, this attack deals four times your normal melee damage. If your check fails, your attack is made with a –2 penalty and doesn’t deal any additional damage. If your strike is a critical hit, you stack the multipliers as normal (PH 304).
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.
Yes, maneuvers can be used with any weapon.
Using a maneuver with unarmed strike is just like using it with any other weapon: it doesn't hurt the user unless the maneuver specifically says so. A shadow replica of your unarmed strike might be a full shadowy replica of the user, or it might be just a floating fist/leg/elbow (depending on how you describe your unarmed attacks).
The associated weapons are part of restrictions in the following feats: Blade Meditation, Desert Wind Dodge, Shadow Blade, Reaping Talons, and Shards of Granite. The Bloodclaw Master prestige class also has an ability that includes a restriction based on a style's associated weapons.
Best Answer
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:
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,
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.