The second part of the question is more straightforward, so I'll look at that first.
Is castling with a "ghost" rook like this legal where there was a rook
on a1 at the start of the game but it was captured before it could
Article 3.8.2 of the FIDE Laws of Chess describes how castling works:
3.8.2 by ‘castling’. This is a move of the king and either rook of the same colour along the player’s first rank, counting as a single move
of the king and executed as follows: the king is transferred from its
original square two squares towards the rook on its original square,
then that rook is transferred to the square the king has just crossed.
This explicitly states that castling involves two pieces, the king and the rook, and it describes how the rook moves in this manoeuvre. If you don't have an unmoved rook on the board then you can't castle. Therefore castling with a "ghost" rook is illegal.
With this clear the first part becomes easier to answer.
Is castling with a "ghost" rook like this legal where there has never
been a rook on a1 because it was given as odds at the start of the
It depends on the rules of the competition or individual game, if the game is a one-off. If the rules state that apart from the initial position of the board the FIDE Laws of Chess apply then clearly it is illegal.
In the absence of such a clarification the best guide is probably the chess historian Edward Winter. In his chess notes for March 2009, note 6029 he considers exactly this question:
6029. Castling with a phantom rook From Mark Thornton (Cambridge, England):
‘In rook-odds games could the odds-giver castle with the “phantom
rook”? For example, if White gave the odds of his queen’s rook, could
he play Ke1-c1? And, if so, were the castling rules the same as if the
rook were present?’
According to Winter there is no conclusive ruling one way or the other. He quotes probably the most sensible approach from Howard Staunton on page 35 of Chess Praxis (London, 1860):
When a player gives the odds of his king’s or queen’s rook, he must
not castle (or, more properly speaking, leap his king) on the side
from whence he takes off the rook, unless before commencing the game
or match he stipulates to have the privilege of so doing.
There are strong feelings on both sides of the argument. Here is more from Winter:
However, in a review of the book on pages 88-89 of the March 1890 BCM
Edward Freeborough disagreed. After stating, with respect to level
games, that castling should be described as a move of the king and
that the king should therefore be moved first, Freeborough observed:
‘It follows logically that the fact of giving the odds of a rook
ought not to deprive the king of his privilege of taking two steps to
the right or left as his first move.’
Page 36 of The British Chess Code (London, 1903) stated:
‘In the absence of agreement to a different effect, a player may
castle (by moving his king as in ordinary castling) on a side from
which, before the commencement of the game, the player’s rook has been
removed, provided that this rook’s square is unoccupied and has been
unoccupied throughout the game, and that the same conditions as to
squares and as to the king are fulfilled which are required for
ordinary castling on this side.’
The above text was quoted on page 275 of the June 1916 Chess Amateur
when a revised edition of the Code was envisaged. Comments were
invited, and on page 305 of the July 1916 issue ‘Simplex’ wrote:
‘This I think sheer nonsense. If a player gave me a rook and wanted
to castle on this rook’s side, I should say, “No, you don’t, you can’t
castle without a castle”. Let’s have no pretence. If a player gives a
rook, let him give it totally not half. Receivers of odds are not
strong players, and to see the nominal giver of odds move his king a
couple of squares would be disconcerting. No; if a player gives odds
let him give them without pretence.’
A contrary view was expressed by W.S. Branch on pages 333-334 of the
August 1916 Chess Amateur:
‘Re Chess Laws, page 305 (July), and as to “castling without rook”, I
would say, first, that you can’t “castle the king” – the full and
proper term, of which “castles” is an abridgment – without a castle.
The phrase should be “moving the king as in castling”. I believe that
the right of the odds-giver to move his king, once in a game, as in
castling, has always been upheld since “castling” was invented
(sixteenth century). It existed, as part of the “king’s leap”, long
before “castling” was invented, and long before the rook was ever
called a “castle”. The giving of the rook as odds should not deprive
the king of any of his rights.’
Branch then gave further historical details regarding the king’s leap.
By 1916, however, the practice of giving odds was disappearing,
without any formal resolution of the ‘phantom rook’ question.
Winter revisits this topic in note 6035. "Castling with a phantom rook (C.N. 6029)" with some interesting anecdotes regarding castling with a ghost or phantom rook.