In Colorado, which is typical of U.S. law, the crime of asking someone to commit a crime is called "criminal solicitation." It is described as follows:
Colorado Revised Statutes ยง 18-2-301. Criminal solicitation (Current
as of January 01, 2022)
(1) Except as to bona fide acts of persons authorized by law to
investigate and detect the commission of offenses by others, a person
is guilty of criminal solicitation if he or she commands, induces,
entreats, or otherwise attempts to persuade another person, or offers
his or her services or another's services to a third person, to commit
a felony, whether as principal or accomplice, with intent to promote
or facilitate the commission of that crime, and under circumstances
strongly corroborative of that intent.
(2) It is a defense to a prosecution under this section that, if the
criminal object were achieved, the defendant would be the sole victim
of the offense or the offense is so defined that his conduct would be
inevitably incident to its commission or he otherwise would not be
guilty under the statute defining the offense or under section
18-1-603 dealing with complicity.
(3) It is no defense to a prosecution under this section that the
person solicited could not be guilty of the offense because of lack of
responsibility or culpability, or other incapacity.
(4) It is an affirmative defense to a prosecution under this section
that the defendant, after soliciting another person to commit a
felony, persuaded him not to do so or otherwise prevented the
commission of the felony, under circumstances manifesting a complete
and voluntary renunciation of the defendant's criminal intent.
(5) Criminal solicitation is subject to the penalties provided for
criminal attempt in section 18-2-101.
Murder is a felony, so it is subject to this statute.
Cases 1, 2, 3, and 5 in the question all clearly come within the scope of this statute. Case 6 does not, because Alice lacks the requisite intent. Case 4 is a close one and a jury would have to decide if Alice's strong implications amounted to an entreaty or attempt to persuade a person, and would have to examine what her intent was in making those implications.
Case 5 is a crime because, while defense of others is a justification for killing someone, the threat must be imminent and deadly defense of others must be the only proportional option, while vigilante justice is not legal. This motive would be a mitigating factor in sentencing, upon conviction, however, and might convince a prosecutor to decline to press charges or to offer a more lenient than usual plea bargain given the circumstances.
Case 6 might still give rise to civil liability, if a jury found in a civil lawsuit, that Alice's actions, while not intentional, amounted to negligent or reckless conduct that caused a wrongful death to be committed, although Alice could not be found civilly liable for conspiracy, which like criminal solicitation, requires actual intent to cause of course of action to be committed.
But, different jurisdictions could define criminal solicitation (or civil liability) in ways relevant to these situations) differently.
The bottom line with respect to the various scenarios presented in the question is that the type of inducement to commit a crime provided isn't really relevant to criminal liability.
Some jurisdictions also make a distinction in the grade of homicide offense that is appropriate (which could influence the severity of the criminal solicitation sentence as well) between "murder for hire" which would include cases 1 and possibly 2, but not the others, and murder for other motives. In jurisdictions where the death penalty is available, murder for hire is often one of the sub-types of murder that is eligible for the death penalty.
