It's not important whether the President can legally do these things, because the rules are so easily circumvented.
In theory the Judicial Branch and the Department of Justice are independent of the executive branch. But in reality the Attorney General (head of the Department of Justice) is appointed by the President, and can be fired at any time if they do something the President disapproves of. In theory the President cannot and should not influence the decisions of the Attorney General, but in reality if they get together in a room where there are no recordings it's virtually impossible to prove it happened. It's also impossible to prevent a President from choosing to appoint only an Attorney General who will do what he says.
There are two checks and balances that are supposed to prevent this (apart from the assumption that Presidents and Attorneys General will behave honourably). Those are 1) the impeachment power of the legislature 2) the ability of the electorate to vote out a President who violates this rule.
President Trump has already signalled his willingness to violate the independence of the Department of Justice by threatening to prosecute his enemies, and to get the cases against himself thrown out. It is virtually certain that his new Attorney General, whoever it is, will abandon all cases against Trump. If that doesn't happen feel free to downvote, but I'm confident that won't be the case.
In the current climate it's hard to imagine that the legislature would impeach and convict Donald Trump for doing those things. And the American electorate have already shown that they are OK with a President who commits criminal acts and interferes with judicial independence. In any case the posts by Alexandria Ocasio-Cortez are intended to state that President Trump will indeed try to do these things, and will probably get away with it, unless there is action by the electorate - such as voting in Senators and Representatives who would in fact impeach him if he did them. Ultimately the responsibility to prevent Presidential abuse of power lies with the electorate.
With thanks to @264champagnebottlesonice, many Republicans promote the "Unitary Executive Theory, in which the Justice Department isn't even theoretically independent of the Executive. They could certainly use this to persuade the legislature - and possibly the courts - that he was justified in directing the Justice Department to do anything he wanted.
The above shows how the President could prosecute his political enemies: and as sfxedit correctly points out, simply prosecuting someone can cause them significant trouble, even if they are eventually exonerated. However the judges (and jury) would decide actual guilt or inocence in a case. But the President also has the power to appoint judges, especially Supreme Court justices who would be expected to ultimately decide a high profile case. Three of the nine are already Trump appointees, and were selected for their views sympathetic to Trump. It is likely that he will appoint more and more judges who agree with his views.
(For anyone thinking this doesn't answer the question, the question was not "Can a President legally do this" it was "Does a President have the power to do this." Those are two different things.)