The president could try this, but anything as blatant as the examples you made would fail to be effective, for many reasons. Lets list out the big ones.
- The president would be removed from office.
This has already been mentioned, so I won't waste too much on it. Congress effectively has the power to remove the president from office for any action they consider sufficiently inappropriate, so long as they can get enough members of the house/senate to agree that what the president did is wrong.
Impeaching a president isn't too hard, since it only requires a majority in the house of representatives, but impeaching is only the first half of the process (basically declaring a trial should be held). However after being impeached to actually remove a president you need 2-3 of the senate to agree. This is very hard to get, in fact it has never happened, though it may have for Nixon if he hadn't resigned first. Since usually you have close to a 50/50 divide between the two major political parties, and generally whatever political party the president is part of will defend him, you will struggle to get much more then 50% of the senate to agree to removal, making 2/3 majority difficult to accomplish. However, if the president is so blatant as to call for murder of someone he doesn't like or anything else so overtly corrupt it's likely even his own political party won't stand by him and we would get our first official removal from office.
- The pardoned individuals will still face trial in state courts
Littering and murder are both crimes in both state and federal courts, but the president only has the power to pardon people of federal crimes. This means that the individual in question are still guilty, and capable of being punished by, the state. This is true for most other crimes you could imagine the president pardoning people for, usually there will also be a law against it on the state level allowing the state to also prosecute the pardoned individual.
In general a presidential pardon will also prevent states from trying the individual because the states are willing to respect the president's pardon, but this policy is not legally required by any laws or the constitution. The state usually, voluntarily, waves it's right to try someone pardoned as part of a general policy that the states and federal governments have to respect each others rulings in difference to double jeopardy; but the state could chose to go against that policy and try someone if they think then pardon was not serving the interest of justice.
To get a little more into the legal weeds to explain why this policy happens you first need to understand that officially, and legally, the constitutional protection against double jeopardy does not protect an individual nearly as well as you may believe. It protects an individual from being tried for the same crime in the same jurisdiction only. In theory a person could be tried by different jurisdictions, such as the state and the federal government. In fact someone could, in theory, be tried for the same crime as many as 7 times
Of course this doesn't feel very just to the average person, and the government agrees. For this reason there has been a standard policy of only allowing a person to be tried once for a crime. If a state tries someone from a crime the federal government wont try him for the same crime, and vice-versa. Nothing legally prevents both parties from doing so, assuming both parties have laws on the books making a crime illegal, but it's generally agreed that in the interest of justice this shouldn't happen. This is a very important policy to both parties, it's held so important that it's only violated in very rare and extreme instances. An example of one of those violations would be during the civil rights movements if someone clearly guilty of a crime was found 'innocent' by a racist state jury the government might also try them because they didn't feel the state racist jury was acting in the interest of justice.
This is relevant because it explains why states usually don't prosecute someone pardoned by the president. It ties very closely with the common agreement not to try someone for the same crime in two jurisdictions, the state treat a presidential pardon similarly to being tried and found innocent and feel that respect for avoiding double jeopardy means they shouldn't try the person in state crimes...usually. Nothing forces this policy, and if the president clearly acted in a blatantly selfish or unjust manner when pardoning the individual of federal crimes the state will likely step in and try the person for the state crime because they feel the federal pardon wasn't in the interest of justice, just as federal government stepped in when people were released due to racism.
- The person will likely be tried on the federal level if the state doesn't step in
I'm working on the presumption that the president would be removed from office if he did anything like encourage a murder by offering to pardon someone. As I already said the most likely outcome is the state would try the person, but if the state didn't step in the federal government may still step in.
A pardon only pardons someone of a specific crime, not of all crimes. This means that someone can still be found guilty of a related crime. For instance a murder may be charged with conspiring with the president to commit murder or assault with a deadly weapon. There is also the common fallback of "conspiracy to deprive an individual of their constitutional right" which is the most common crime the federal government uses when they think that something happened to make a state trial not be sufficient (such as when racist jury set a guilty man free). The federal government can still charge the individual with these, and likely many other crimes, that he was also guilty of when he killed someone, and since the president did not originally pardon the man of these crimes, and is no longer in office to do so when the man is arrested for the new charges, the murderer will still face trial in federal court for them.
- A rational man wouldn't risk a pardon
For the reasons listed above, plus the need to trust that a president would risk loosing his position by abusing a pardon and not be removed from office before he can act, there is a very real chance that a murderer will still end up spending his life in jail even if the president promises to pardon him. As such a wise person will not commit the murder even with a promise of a pardon, though most criminals are not acting very rationally (many crimes do not offer sufficient benefit to justify the risk of jail time when considered from purely rational level) so he may still find someone crazy or stupid enough to commit the crime.
All of these examples assume the president were to blatantly abuse his powers as president. In most of the cases someone has to do something that breaks tradition, such as removing a president or violating a very sacred policy of avoiding double jeopardy across jurisdictions, to stop the president. If a president were to act in a blatantly immoral manner people will step up to resist him, but what if the president's actions weren't quite so obviously wrong?
In fact presidents have effectively overruled laws by pardoning individuals found guilty of them, or simply putting out a presidential order to not enforce the law, on numerous occasions. K dog gave one example, but I think an even more obvious example in the modern era would be marijuana use.
Use, and ownership, of marijuana is technically illegal on the federal level, despite being made legal in a number of states. The reason that people are able to openly use, and sell, marijuana despite it being illegal federally is that Obama basically put out a presidential order stating that we should not prosecute anyone in those states for marijuana use. Effectively Obama invalidated the federal law to allow the states to try experimenting with marijuana. Now that protection only was guaranteed whiles while Obama is in office, Trump could easily have changed that policy, as could the next president, though for now Trump seems fine with allowing that policy to stay in place.
So a president can invalidate laws with pardons or, more often, executive orders. However, he generally needs some level of justifiably to do it. He needs at least a sizable minority of individuals, usually those in his party who are more likely to defend him in principle, to side with him. If he ever abuses this power too far there are some checks and balances, both officially and unofficially, which could step in to stop him.