A TLS handshake concludes with unilateral authentication of the server. For (EC)DHE-only key exchange, the server sends a Certificate
message followed by a CertificateVerify
message, the former contains a certificate (along with its certificate chain) for authentication, and the latter contains a signature (constructed with the private key corresponding to the public key in the certificate) over a hash of the handshake's transcript, thereby, proving possession of the private key used for signing, hence, identifying the server.
The certificate chain comprises DER-encoded X.509v3 certificates (unless an alternative was negotiated). The server's certificate must appear first and every subsequent certificate should certify the previous one (i.e., every subsequent certificate should contain a signature - using the private key corresponding to the certificate's public key - over the previous certificate's public key), hence, the list is a certificate chain.
Regarding your questions, you're creating (q2) and verifying (q3) the certificate.
Further details: https://arxiv.org/abs/1904.02148 (Section 2.5).