That would be up to the Prime Minister and his/her perception of potential public reactions. But presumably he/she would offer the life peerage precisely to be a member of the House of Lords, so to ask for a hereditary title would probably not be well received.
Anyway IIRC the last hereditary titles (outside the royal family) were Thatcher’s nominations, one of which was the earldom for the former PM Macmillan (as had previously been traditional for former PMs), but the only others went to her deputy (who had no male heir anyway) and a baronetcy (=a hereditary knighthood, so less than a peerage) to her husband - he might well be said to have earned it by that simple fact, but it also meant that the title would (as it did) pass to their son, the unspeakable Mark.
So as a practical political reality, the answer would be “On your bike”.
You mean: What would it take to change the rule that a peerage is extinguished when it is merged with the Crown (i.e., the holder becomes King)? That’s a common law rule, so it would, again, require an Act of Parliament. The newly ascended King could not change it to retain the title because he cannot legislate to change the common law. I don’t think he could grant himself the title again.
But note that the Duchy of Lancaster is exempt from this rule. The reigning British monarch is also, in personal union, Duke of Lancaster.
Maybe not. The reigning monarch is entitled to the income from the Duchy of Lancaster, and as a matter of tradition has often used the title of Duke. However, in the Case of the Duchy of Lancaster (1561), the judges ruled that Elizabeth I held the Duchy as Queen rather than as Duke (or Duchess). It’s a complicated issue, and apparently George V was given legal advice that he was almost certainly not the Duke of Lancaster (but he also approved a loyal toast for use in Lancashire that included the ducal title).