Question 1: Why is that one(in space) considered alpha and not the carbon atom next to it?
All human proteins consist of $\alpha$-amino acid residues. An $\alpha$-amino acid means the carboxylic acid group ($\ce{COOH}$) and amino group ($\ce{NH2}$) are separated by one $\ce{C}$ carbom atom, which is called $\alpha$-carbon ($\ce{C}_\alpha$; See the insert at bottom right of the diagram):

Usually, backbone of a protein ($\alpha$-helix) is written as: $$\ce{H2N-C_\alpha(R^1)-C(=O)-NH-C_\alpha(R^2)-C(=O)-NH-C_\alpha(R^3)-C(=O) -}\cdot \cdot \cdot \ce{-NH-C_\alpha(R^n) -COOH}$$
For example, the dipeptide in this diagram can be written as:
$$\ce{H2N-C_\alpha(R^1)-C(=O)-NH-C_\alpha(R^2) -COOH}$$
As demonstrated in these backbones, you'd see each $\ce{C_\alpha(R^1)}$ is in place between $\ce{NH}$ and $\ce{C(=O)}$ (the $\ce{R^1, R^2,}$ etc. are corresponding side chains of the particular amino acids). Thus, $\text{<Atom C>}$ next to $\text{<Atom CA>}$ ($\ce{C}_\alpha$) in the written chart of the program is referring to the carbonyl $\ce{C}$ of that particular amino acid. For example, let's consider three amino acids in the given chart: (10 Arginine; (2) Lysine; and (3) Leucine:

Arginine (arg) residue in the chart is can be written as follows (the side chain in parenthesis):
$$\ce{-HN-C_\alpha(CH2CH2CH2NHC(=NH)NH2)-C(=O) -}$$
According to the nomenclature off the program, you can rewrite it as:
$$\ce{-HN-C^A(C^BH2C^GH2C^DH2N^EHC^Z(=N^{H1}H)N^{H2}H2)-C(=O) -}$$
where $\ce{A #} \alpha$; $\ce{B #} \beta$; $\ce{G #} \gamma$; $\ce{D #} \delta$; $\ce{E #} \epsilon$;$\ce{Z #} \zeta$; and $\ce{H #} \eta$ (following Greek alphabet letters). Since two nitrogen atoms are attached to $\ce{C}_\zeta$, they are appropriately labelled as $\ce{N^{H1}}$ and $\ce{N^{H2}}$ after next Greek letter $\eta$.
Since the program has avoided hydrogen, let's rewrite it again without hydrogen atoms:
$$\ce{-N-C^A(C^BC^GC^DN^EC^Z(=N^{H1})N^{H2})-C(=O) -}$$
Thus, program writes it as $\ce{N -C_\alpha -C(=O) -}$ first and then the the atoms in side chain next. Hence, Residue ARG: $\text{<Atom N>}$, $\text{<Atom CA>}$, $\text{<Atom C>}$, $\text{<Atom O>}$, and then the side chain in parenthesis as: $\text{<Atom CB>}$, $\text{<Atom CG>}$, $\text{<Atom CD>}$, $\text{<Atom NE>}$, $\text{<Atom CZ>}$, $\text{<Atom NH1>}$, $\text{<Atom NH2>}$.
Similarly, lysine (lys) residue in the chart is (side chain in parenthesis):
$$\ce{-HN-C_\alpha(CH2CH2CH2CH2NH2)-C(=O) -}$$
You can rewrite it according to the nomenclature off the program (avoiding $\ce{H}$s):
$$\ce{-HN-C^A(C^BC^GC^DC^EN^Z)-C(=O) -}$$
Thus, program writes it as $\ce{N -C_\alpha -C(=O) -}$ first again, followed by the atoms in side chain. Hence, Residue LYS: $\text{<Atom N>}$, $\text{<Atom CA>}$, $\text{<Atom C>}$, $\text{<Atom O>}$, and then the side chain in parenthesis as: $\text{<Atom CB>}$, $\text{<Atom CG>}$, $\text{<Atom CD>}$, $\text{<Atom CE>}$, $\text{<Atom NZ>}$.
For leucine, $\ce{-HN-C_\alpha(CH2CH(CH3)CH3)-C(=O) -}$,you can again rewrite the formula according to the nomenclature off the program (avoiding $\ce{H}$s):
$$\ce{-N-C^A_\alpha(C^BC^G(C^{D1})C^{D2})-C(=O) -}$$
Note that since two carbon atoms are attached to $\ce{C}_\gamma$, they are appropriately labelled as $\ce{C^{D1}}$ and $\ce{C^{D2}}$ after next Greek letter $\delta$.
Thus, program writes it as $\ce{N -C_\alpha -C(=O) -}$ first again, followed by the atoms in side chain. Hence, Residue LEU: $\text{<Atom N>}$, $\text{<Atom CA>}$, $\text{<Atom C>}$, $\text{<Atom O>}$, and then the side chain in parenthesis as: $\text{<Atom CB>}$, $\text{<Atom CG>}$, $\text{<Atom CD1>}$, $\text{<Atom CD2>}$.
Question 2: Is it a given that each residue out there has only one alpha-carbon?
As explain in above backbone of the protein, you'd find only one $\ce{C_\alpha}$ fir each amino acid (which is chiral).