Prince Christian (X) represented his grandfather Christian IX at the baptism.
Yes, at least four times:
1868 - after the birth of Nicholas (II).
1881 - after Alexander III's accession to the throne.
1891 - celebration of Dagmar and Alexander's silver wedding.
1894 - funeral of Alexander III and wedding of Nicholas II.
Yes, for instance in 1871, but I don't know how often.