Is the α-carbon considered a part of amino acid side chain?
Is the nitrogen atom on the backbone is considered a part of the side chain of the proline?
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This depends on the context.
In a context where atoms are either classified as backbone or as sidechain, you would count the atoms C, O, N, CA (and connected hydrogens) as backbone, and all the other atoms as sidechain. This is how molecular viewers classify atoms.
(Jmol image of Pro22 and Glu23 of crambin, 1CRN, with sidechain atoms shown as spheres.)
According to IUPAC/IUB's JCBN,
2.2.5. Definition of Side Chain
When amino acids are combined in proteins and peptides, C-1 , C-2 and N-2 of each residue (the numbering being that of aliphatic amino acids) form the repeating unit of the main chain ('backbone') and the remainder forms a 'side chain'. Hence the words 'side chain' refer to C-3 and higher numbered carbon atoms_ and their substituents.
If you show the structure of a sidechain, it is highly likely you would include at least the alpha carbon for context, and in case of proline the backbone nitrogen as well to illustrate the ring structure.