By default, it's the VP + a majority of the cabinet, but Congress can pass a law substituting another group or commission for the cabinet. In either case, the President can appeal to Congress if he disagrees, in which case it requires a 2/3 majority in both houses to uphold the decision that the President is incapacitated.
Exact relevant language of the amendment:
Section 4. Whenever the Vice President and a majority of either the principal officers of the executive departments or of such other body as Congress may by law provide, transmit to the President pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office, the Vice President shall immediately assume the powers and duties of the office as Acting President.
Thereafter, when the President transmits to the President pro tempore of the Senate and the Speaker of the House of Representatives his written declaration that no inability exists, he shall resume the powers and duties of his office unless the Vice President and a majority of either the principal officers of the executive department or of such other body as Congress may by law provide, transmit within four days to the President pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office. Thereupon Congress shall decide the issue, assembling within forty-eight hours for that purpose if not in session. If the Congress, within twenty-one days after receipt of the latter written declaration, or, if Congress is not in session, within twenty-one days after Congress is required to assemble, determines by two-thirds vote of both Houses that the President is unable to discharge the powers and duties of his office, the Vice President shall continue to discharge the same as Acting President; otherwise, the President shall resume the powers and duties of his office.
Naturally, a comatose President would have difficulty telling Congress that he's fit for duty.
To the best of my knowledge, Congress has not passed a law replacing the Cabinet as the deciding body, but may have passed a law clarifying the composition of the Cabinet for 25th amendment purposes.
To answer the OP's other questions, yes, the VP is Acting President only in this case. Impeachment is the only Constitutional mechanism for removing a President before his term is up, other than death or resignation, and impeachment probably isn't appropriate since it's hard to argue that being in a coma constitutes a high crime or misdemeanor.
The Constitution does not define "death". I'd expect that Congress could by statute clarify the standards and process for declaring the President dead, so long as the standards and process are within the bounds of common usage of the word "dead" (e.g. brain death could probably be included in the definition, but a law that declares the President dead next time he takes a nap would be unconstitutional). In the absence of such a law, general prevailing practices for declaring death would probably be accepted (for example, if he has a stroke and is taken to a hospital in Virginia, and a doctor there declares death in a manner consistent with Virginia law and practice, then the President is dead).
The legal interpretation could get a bit hairy if it wound up in court somehow, though, since there's conflicting theories about textual interpretation of the Constitution. What I just described is a bit of a loose construction, looking at Constitutional language in light of current usage and practice. A strict textualist/originalist interpretation would look at the generally accepted meaning of the words at the time the amendment was proposed and ratified, so brain death would only count if "dead" was generally understood to include brain death in the mid-1960s. On the other extreme, a "living document" standard would take a more pragmatic approach and interpret any ambiguity in the text in the way that best achieves a just, reasonable outcome by currently prevailing standards, which probably would count irreversible brain death as "death".
By any standard, though, the President would not be considered dead on the basis of "probably will never recover". I'd need to be beyond reasonable dispute that the President's down for the count.