These Ruling(s) have not been confirmed as applicable to the TCG.[Notes 1]
Upon entering his Main Phase, the turn player has Priority to perform the first action. Player B will only be able to activate "Mind Crush" first if the turn player chooses to not perform an action and passes Priority.
A player is allowed to declare any card in the TCG as a card for Mind Crush. This would include a card that is in the Forbidden List. Hand checks for the opponent is only allowed if all legal copies of the cards are not accounted for in all Public Knowledge areas. Since the legal total is 0, you will not get to verify your opponent's hand if you call a Forbidden card or if all legal copies are in a Public Knowledge area.
If the maximum legally allowed number of copies of the declared card cannot be verified as public knowledge, you may ask your opponent to verify their hand.
In place of declaring a FULL card name, a precise description of a card's stats, appearance, and/or effects, is considered acceptable so long as both players are in clear agreement over which card is being described.
In the case of a card like "Luster Dragon", since the original printing of "Luster Dragon #2" did not have the #2 by its card name, you should ask the player for additional information. If they say "The Level 6 one" is then enough information provided so that both players know which card it is.
If a card must be shown to the opponent, like by the effect of "Mind Crush", "Trap Dustshoot", or by a "search effect" like "Sangan", it becomes face up, it is considered public knowledge, and as such the opponent is entitled to read the card. The opponent should do this in a timely manner, and not delay the game.
Both "Rai-Mei" and "Raimei" have their names pronounced "raimei", so if you wish to declare either card with "Mind Crush", then you must further describe the card to be declared until your opponent understands which card you are declaring. For example, you could declare "the Monster Card raimei" or "the Spell Card raimei", etc.
These TCG rulings were issued by Upper Deck Entertainment and have since been deemed unofficial by Konami. They were previously considered official and can still be correct, as long as they follow the official gameplay rules, or unless Konami has issued a ruling that says otherwise. References: 
↑Konami Judge Program Rulings are considered unverified, as their source(s) are not publicly viewable. They are not confirmed as applicable to the TCG until they have been officially published by Konami.
↑Konami FAQ: When you declare the card name of "Rai-Mei" or "Raimei", is it necessary to distinguish between the two?
↑Konami OCG FAQ: As for "Mind Crush", upon declaring a card name, it is still uncertain whether you or the opponent will discard a card. In this case, can the effect of "Gravekeeper's Watcher" be chained to the activation of that "Mind Crush"?