Are these proofs logically equivalent? Here are two proofs, firstly:
x = 0.999...
10x = 9.999...
    = 9 + 0.999...
    = 9 + x
9x = 9
x = 1

And secondly:
x = 1 - 1 + 1 - 1 + 1 - 1 ...
  = 1 - (1 - 1 + 1 - 1 + 1 ...
  = 1 - x
2x = 1
x = 1/2

The fourth line in the first and the third line in the second use the same trick. Does the legitimacy of the first proof guarantee that of the second?

I would say no.
0.999.. is a symbol for a rational number. So for example, $\frac{1}{3}$ is by definition the number with the property
$$ \frac{1}{3} \cdot 3 = 1,$$
and $\frac{1}{3}$ is just a symbol that we use for this number with this property. Another symbol of the same number is also
$ 0.333... $ All operations on the first proof are justified.
For the second proof $1-1+1-1 ....$ is no real number. It is not clear what you mean with this symbol, and, therefore, you can not add something to some strange symbol that has no meaning.
