Separation of powers is a guideline, or perhaps it might better be called a design goal, of the US federal government, not an unbreachable rule. Strictly speaking, confirmation of executive appointments by the Senate is a violation of the separation of powers. Read The Federalist for a detailed explanation of the complex way in which powers of the three main departments are separated in some matters and intermixed in others, and of the reasons why the thee authors, all key members of the convention that drafted the Constitution, thought this wise.
Specifically in No 51 of The Federalist, it is stated:
In order to lay a due foundation for that separate and distinct exercise of the different powers of government, which to a certain extent is admitted on all hands to be essential to the preservation of liberty, it is evident that each department should have a will of its own; and consequently should be so constituted that the members of each should have as little agency as possible in the appointment of the members of the others. Were this principle rigorously adhered to, it would require that all the appointments for the supreme executive, legislative, and judiciary magistracies should be drawn from the same fountain of authority, the people, through channels having no communication whatever with one another. Perhaps such a plan of constructing the several departments would be less difficult in practice than it may in contemplation appear. Some difficulties, however, and some additional expense would attend the execution of it. Some deviations, therefore, from the principle must be admitted. In the constitution of the judiciary department in particular, it might be inexpedient to insist rigorously on the principle: first, because peculiar qualifications being essential in the members, the primary consideration ought to be to select that mode of choice which best secures these qualifications; secondly, because the permanent tenure by which the appointments are held in that department, must soon destroy all sense of dependence on the authority conferring them. [Emphasis added.]
The Wikipedia article about this essay states:
Federalist No. 51 addresses means by which appropriate checks and balances can be created in government and also advocates a separation of powers within the national government. The idea of checks and balances is a crucial part of the modern U.S. system of government.
As to the specific questions:
In the unlikely event that a US President nominated the sitting vice-President to the US Supreme Court (it has never been done), if the senate split 50-50 on confirming such a nomination, the VP could indeed vote to confirm that nomination, although for political reasons, the VP might abstain (and a tie vote loses in the Senate).
If no one person has 270 electoral votes for VP, the senate chooses from among the top two electoral vote getters for VP. If the senate split equally between those two, the then-sitting VP could vote to break the tie.
If and only if such a VP was one of the top 2 (presumably if running for reelection), then such a VP could vote for himself or herself. I would not call a vote to choose a person as VP who was nominated, was one of the top-2 electoral vote getters, and was the recipient of the votes of 50 senators a "power grab". Someone must be chosen to be the new VP. If 51 Senators cannot agree, then the sitting VP must break the tie.
In any case, that is how the Constitution, as currently amended, now reads. It could be changed, the process for electing the VP was changed once, by the 12th amendment. Whether it should be changed, to handle this case which has never yet occurred, is a matter of politics, not law.