The way it works in the US, and many other federations, is that officially, the federal government has certain enumerated powers. Some of them are very broad, some less so. All powers not enumerated as federal are state powers that the federal government cannot touch.
You give the following specific examples. I am not a lawyer and am not even from the US, although I have some interest in its legal system, so please do not treat me as an authority on any of this.
You write that the federal government can overturn state abortion bans. You are probably thinking of the proposed Women's Health Protection Act. If you read that bill, you can see that its sponsors believe that this is a federal power because of:
the powers of Congress under the commerce clause of section 8 of article I of the Constitution of the United States, its powers under section 5 of the Fourteenth Amendment to the Constitution of the United States to enforce the provisions of section 1 of the Fourteenth Amendment, and its powers under the necessary and proper clause of section 8 of article I of the Constitution of the United States.
At least the citation of the commerce clause certainly makes some sense: healthcare services are a form of commerce, and commerce "with foreign Nations, and among the several States, and with the Indian Tribes" (and that has been very broadly interpreted because services available in one state can also affect interstate commerce) is a federal power.
I do not think the federal government could do that in the US. Some people might argue that compulsory education is a form of "involuntary servitude" that is (or may be) prohibited under the Thirteenth Amendment (which also contains a federal power to enforce by appropriate legislation); but reading through this Wikipedia section, it seems unlikely that that would hold up in court.
Lower the age to vote in state elections
Certainly so if it is over 18 years; the 26th Amendment says that Congress has the power to enforce it by appropriate legislation. Otherwise I do not think that can be read into any enumerated federal power.