Does knight behave like a king in his infinite odyssey?

Consider the following open knight's tour on a $5\times 5$ board, starting at position $1$ and then touring the $5\times 5$ board in the indicated move order. The final position is $25$, from which the knight can exit at either $A$ or $B$.

                         A

     5   22   17   12    7    B

    16   11    6   25   18

    21    4   23    8   13

    10   15    2   19   24

C    3   20    9   14    1

     D

So the tour starts at a corner, and can exit in either direction at an adjacent corner. By a suitable reflection of the same tour, we could also arrange to exit at C or D, if desired.

Therefore, with this pattern and its copies, we can proceed to tile any $5\times 5$ board and then arrange so as to enter any of the four adjacent $5\times 5$ boards on a corner as desired.

This allows us to tile any sequence of $5\times 5$ boards that are each connected to the next by a common edge. In particular, we can carry out your spiral pattern, since in that pattern each block has a common edge with the next block.

Thus, the answer is affirmative, there is a knight's tour of the infinite $\mathbb{Z}\times\mathbb{Z}$ chessboard.

Indeed, since this method allows one to follow any tiling of the plane by $5\times 5$ blocks, where each block is attached to a next block by an edge, it follows that there are continuum such knight's tours.

(Moved from a comment.)

Questions 2, 3, and 4 are answered negatively by [W], which decomposes the plane into concentric annuli of width two.

OP observes that this construction is still spiral (every move after the first rotates the direction vector from the origin to the knight clockwise). However, note that the move to transition from one ring to the next could be made so as to reverse the direction of the spiral. Instead of ending an annulus by moving (1,-2), the knight could move (2,-1) and fill the next annulus moving always counterclockwise. (Of course, the king could double-back the same way in its spiral tour.)

[W] Warendorff, Jay, "An Infinite Knight's Tour", the Wolfram Demonstrations Project, http://demonstrations.wolfram.com/AnInfiniteKnightsTour/ .

I've pictured the start of a more-or-less random-5-by-5 Hamkins-variant infinite Knight's tour construction. It begins at the center and proceeds to the SE corner of the 25-by-25 board.