Hypothetically, could the U.S. Congress abolish itself and transfer all power to the President? I'm thinking of a scenario where Congress amends the Constitution and gives all power the President, while also removing its own power.
Theoretically Yes, but they couldn't do it on their own
Amendments to the US Constitution can change or overrule earlier parts without issue. For example, the 17th Amendment modified the part of Article 1, Section 3 which said that Senators were elected by the State Legislature, and replaced it with direct elections by voters in each state.
Therefore, there is no reason that a constitutional amendment couldn't make other fundamental changes, such as letting the President appoint Congress, or giving the President the ability to propose and approve bills.
However, even if they wanted to give up their power, Congress couldn't pass a Constitutional Amendment on their own. Article 5 of the US Constitution dictates a 2-step process for passing an amendment:
The Congress, whenever two thirds of both Houses shall deem it necessary, shall propose Amendments to this Constitution, or, on the Application of the Legislatures of two thirds of the several States, shall call a Convention for proposing Amendments, which, in either Case, shall be valid to all Intents and Purposes, as Part of this Constitution, when ratified by the Legislatures of three fourths of the several States, or by Conventions in three fourths thereof, as the one or the other Mode of Ratification may be proposed by the Congress;
First, an amendment must be proposed, either by passing both the House and the Senate with a 2/3 majority, or by being approved at a Constitutional Convention called by the states (which has never happened, after the original Convention).
Then, the amendment must be ratified by the State Legislature, or a state Constitutional Convention, in 3/4 (currently 38) states.
So, it wouldn't be enough for Congress to want to institute a dictatorship, the individual states would have to agree too.
There's one other limitation:
The last clause of Article 5 puts another limitation on constitutional amendments:
Provided that ... no State, without its Consent, shall be deprived of its equal Suffrage in the Senate
You could make an argument that abolishing the Senate would maintain equal Suffrage (every state gets the same 0 Senators), but any State that opposed this plan would almost certainly have grounds to sue to the Supreme Court on these grounds.
Hypothetically, could the U.S. Congress abolish itself and transfer all power to the President? I'm thinking of a scenario where Congress amends the Constitution and gives all power the President, while also removing it's own power.
Congress can proposed amendments to the U.S. Constitution, but it can't adopt amendments to the U.S. Constitution without approval from state legislatures. So, because Congress can't amend the Constitution, the answer is no.
Furthermore, Article V of the U.S. Constitution provides that:
no State, without its Consent, shall be deprived of its equal Suffrage in the Senate
via a Constitutional Amendment. And, I do not believe that any fair reading of the U.S. Constitution would find that abolishing the U.S. Senate entirely would not violate this provision.
Since this is beyond the scope of a permitted constitutional amendment, the only way to secure your proposed change would be to violate the law and the constitution in some sort of coup or revolution (bloodless or otherwise) and to establish a new regime, much as the United States did when it extra-legally declared its independence from the British Monarch.
Congress also cannot pass a law that would do this because the non-delegation doctrine (a separation of powers concept that is part of U.S. Constitutional law) provides that a law that does not provide an "intelligible principle" to govern how it should be carried out is unconstitutional and void.
Congress can't do it alone. Congress could start the process by sending the necessary constitutional amendment for ratification to the states. But it would only work if every state ratified it.
Why? Because the constitution requires that states not be deprived of equal suffrage in the Senate without the consent of each state so deprived. So unlike a normal amendment that requires ratification by three quarters of the states, this would only take effect if every state consented. It seems reasonable that ratification by the state would signal the state's consent. Although we've never reduced a state's suffrage in the Senate, so the process is not exactly well developed.
It could also work the way that adopting the constitution in the first place did. The states left the original government and joined the new one. Under the current constitution, that could start happening with only three quarters of the states ratifying. So the remaining states would have equal suffrage in the smaller remaining Senate.
So this is not entirely impossible (except that Congress can't do it alone). But neither would it be an easy thing to do.
Note that it would actually be easier for the states to do it. The states don't need Congress to pass an amendment, whereas Congress does need the help of the states. Assuming unanimous agreement among the states, they could call a constitutional convention with only two thirds of the states. Then they could ratify with only three quarters. Then it just remains to get unanimous consent to the elimination of the Senate (the House can go at three quarters). But all this assumes a unanimous decision of the states.
No president since James Monroe has received a unanimous vote from the states much less anything close to support for a permanent dictatorship. Only George Washington ever got every electoral college vote. Source: Wikipedia.
Even really popular modern presidents like Franklin Delano Roosevelt in 1936 (Maine and Vermont), Richard Nixon in 1972 (Massachusetts), and Ronald Reagan in 1984 (Minnesota) lost at least one state.