You’re correct that in an alpha helix, the hydrogen bonds occur within the same polypeptide chain. The hydrogen bond forms between the carbonyl oxygen of one amino acid and the amino hydrogen of an amino acid four residues earlier, which gives the helix its characteristic shape.
Beta sheets also involve hydrogen bonds, but they can occur either within a single polypeptide chain or between different chains. In a beta sheet, the hydrogen bonds form between the carbonyl oxygen and amino hydrogen of amino acids on adjacent strands, which can be on the same chain (if it loops back on itself in a hairpin) or on different chains.
To differentiate between parallel and antiparallel beta sheets, you’ll look at the direction of the peptide chains. In an antiparallel beta sheet, adjacent chains run in opposite directions, so the N-terminus (amino end) of one strand is next to the C-terminus (carboxy end) of the neighboring strand. The hydrogen bonds in an antiparallel sheet are directly across from each other, leading to a more linear and stronger bond. In a parallel beta sheet, adjacent chains run in the same direction, so the N-terminus of one strand is near the N-terminus of the next strand and the same for the C-termini. The hydrogen bonds in a parallel sheet are offset, which leads to a less linear and therefore slightly weaker bond.
The side chains (R groups) in a beta sheet alternate above and below the plane of the sheet, regardless of whether it’s parallel or antiparallel. So while looking at the direction of the R groups can be helpful, it won’t tell you whether the sheet is parallel or antiparallel.
The kink in beta sheets and the twist in alpha helices are both due to the physical properties of the polypeptide chain and the way that the amino acids interact with each other.
In the case of beta sheets, a kink can occur when there’s a proline residue. Proline has a unique structure among the amino acids, with its side chain forming a ring with the backbone. This makes it less flexible and unable to fit into the regular alignment of a beta sheet, causing a kink.
The twist in an alpha helix is due to the geometry of the amino acids and the way they pack together. Each amino acid is rotated slightly relative to the previous one, and over the length of the helix, this results in a twist. The rotation is driven by the hydrogen bonds between the backbone atoms: the formation of these bonds is energetically favorable, and the amino acids rotate to allow as many bonds to form as possible.