No, they are not the same.
The sequence produced by cut-and-project to line $y=\frac x\phi$, named "the CP chain" for brevity, is not the Fibonacci chain.
$LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS\cdots$ is the Fibonacci chain.
$LSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSL\cdots$ is the CP chain obtained in the question, where the first letter, $L$ corresponds to the first segment on that line starting from $(0,0)$ up-rightwards and the second letter, $S$ corresponds to the second segment, etc.
The two chains differ at position $4,5,12,13,25,26,33,34, ...$
Not eventually the same, either
For ease of description, use $0$ and $1$ instead of "L" and "S". The Fibonacci chain is $0100101001001010010100100101001001\cdots$ while the CP chain is $0101001001010010010100101001001010\cdots$.
The $n$-th digit of the Fibonacci chain is, as given here,
$$f_n=2+\lfloor n\phi\rfloor -\lfloor(n+1)\phi\rfloor$$
The $n$-th digit of the CP chain is
$$c_n=2+\lfloor (n-1)\phi + \frac12\rfloor -\lfloor n\phi+\frac12\rfloor$$
Since $\phi$ is irrational and $\frac12$ is rational, it can be proved that the Fibonacci chain and CP chain are not eventually the same in the sense that there are no integer $n_f$ and $n_c$ such that $f_{n_f+n} = c_{n_c+n}$ for all $n\ge0$.
Fibonacci chain can be obtained by cut-and-project method.
On the other hand, the Fibonacci chain and the CP chain share many properties, some of which are explained in the paper mentioned in the question.
The Fibonacci chain can be obtained by the cut-and-project method, in fact. In stead of line $y=\frac x\phi$, use line $y=\frac x\phi +(1-\frac{\phi}2)$. The segments obtained on the new line starting from $(\frac{4-3\phi}{10}, \frac{3-\phi}{10})$ up-rightwards will be encoded to the Fibonacci chain.
There are plenty of counterexamples to your conjecture $(2)$. If I'm honest, I don't think there's much chance of rescuing it either. There's not really much about aperiodicity that relies on some underlying irrationality of the method for producing the tiling. Of course, the converse is generally true - irrationality, say from the expansion factor of a substitution rule, or the slope of a cut-and-project generating method, will imply aperiodicity.
To give a concrete example, the two-dimensional Thue-Morse substitution gives rise to an aperiodic (repetitive) tiling with linear expansion factor $\lambda = 2$. This comes from taking the product of the Thue-Morse substitution with itself and suitably colouring the symbols. The substitution rule is as follows (the rule for the white tile is given by just inverting all colours in the rule for the black tile):
[image taken from https://arxiv.org/abs/2302.12908]
Moreover, unlike for the chair tiling, this tiling is not limit-periodic, as the 2d Thue-Morse substitution is bijective, and so the tiling dynamical system does not have discrete spectrum.
Best Answer
We may apply the same cut-and-project technique to any 3D lattice and a plane. Obviously, if the plane has irrational slopes to the axes, the resulting pattern will not be periodic.
The nature of the tiles is another story altogether. Here, for example, we only have 3 different tiles, and they may produce all kinds of patterns, some periodic and some not.