More inspiring spirals (1)

(click here to see the motion)

Hello Math-Fun,

Pick at random an integer k on this (expanding to the infinite) square grid: there is always at least one of the digits of k present in one of its 8 immediate neighbors.

Example:

If you look at 31, in the upper left corner, you will see that 31 shares a 3 with 63, and (clockwise from there) a 1 with 123, 117, 51, 16, 17, 115 and 116.

How was this grid build?

Start with a 0 (zero) somewhere and fill the grid with the lexicographically earliest infinite square spiral of distinct nonnegative integers that don't lead to a contradiction.

The next term after 0 must be 10, as 10 is a neighbor of 0 and the smallest available integer containing the digit 0; after 10 we must take 20 for the same reason (20 is a neighbor of 0 and the smallest available integer containing 0). We go on spiraling like this until 80.

The next available integer is 18 (and not 90) – why? Because 18 has only 2 immediate neighbors yet ; 80 and 10. As 18 shares a digit 8 with 80 and a digit 1 with 10, we keep 18 (instead of 81 or 90 or infinitely many others like 1080).

What comes next? 100, no less (see below) – as 100 shares at least one of its digits with 20, 10, 80 and 18, which are the neighbors so far of 100.

We extend, after 100, the spiral with the integers 12, 2, 23, 102 (...) and 11:

The above green digits 1 and 2 are "reminders": «Dear computer, remember there must be a digit 1 in the cell above 108, as this cell is an immediate neighbor of the integer 1 (already in the grid); there must be a digit 1 in the cell above this integer 1 for the same reason, (...) and there must be a digit 1 in the cell under 11 too» – the same with the green digits 2. 

The spiral is thus extended hereunder according to the constraints (the integer 22 will produce a few "reminders" too):

The next "reminders" are given by the integer 9, hereunder:

And so on – the grid below is now the same as the first one:

Questions:

Is the spiral a permutation of the nonnegative integers?

If this is true, what could be the index of the integer 3 in the spiral?

(caveat, grid computed by hand – typos and errors almost guaranteed)