In Russia or elsewhere, was it normal to get a surname at baptism?
Did any rules guide selection of the surnames?
What sorts of surnames were used other than the officiator's own?
For example, in the territory of modern-day Republic of Latvia, surnames arrived gradually and in different ways:
[Note, as I am no historian I may mix up some of the terms such as "peasant", "farmer", "free tenant", especially since in Latvian they are sometimes also used in place of each other with the single name "zemnieks"(farmer). Please indicate corrections, where appropriate.] The following is translation from Latvian:
The first surnames appeared with German entry into Livonia. Initially only German baronial families had them, but later on also city-dwellers and free farmers (free peasants?). Serfs didn't have surnames. Usually serfs were identified by the name of their house, traits of character or occupation, and within borders of a single manor that was enough.
At the time of emancipation of serfs (after 1819), farmers could move freely between manors, and the problem of personal identification and entries in manor documents was raised. Therefore farmers were ordered to adopt surnames. The first "surnamings" took place in Vidzeme (part of Latvia) from 1822-1826.
The surname could be adopted only by the elder of the family (dzimta, in these terms a bit broader term than nuclear family of today), and the same surname had to be used by his sons and grandsons. If any of the sons had started an independent life, he could take his own surname. Brothers could take each their own surnames also if the father (the eldest of the family) was dead.
So as you can see, no direct linking to baptism -- territory of Latvia was Christianized centuries(12th-14th century) before surnames were adopted.
Choice of surname
The peasant had to choose surname by himself, and it was forbidden for it to be inappropriate to his class: he couldn't take a surname of German gentry, their family or surname of a famous person/family. It was advised to choose Latvian surnames.
The main types of surnames were based on:
Place names: name of house or closest place
Countryside and nature: Eg. Putniņš (Birdie), Žagata (Magpie), Ozoliņš(Oak-ie), Krūmiņš (Bush-ie) [-ie being my attempted translation of the denuminative form of -iņš)
Foreign names or father's name: Jēkabsons, Pētersons, Neilands, Lembergs…
Profession and occupation: Kalējs (Smith), Mūrnieks (Mason)
Double surnames: Dauge-Daugava, Rieksta-Riekstiņa
Human characteristics: Strups (Short), Resnis (Fat), Zilgalvis (Blue-head)
Human names: Valters (Walter), Miķelis (Michael)
Nationalities: Krievs (Russian), Lībis, Letis (Lett = Latvian)
Orthodox church members and Russians used also the patronymic.
Source: Blog of Latvian State History Archive, which in turn refers to a journal publication:
Andrejs Plakans, Charles Wetherell: Patrilinear geneology, surnames and family identity: Baltic governorate of the Russian Empire in 19th century, published in "Latvian Archives", 2003, #3. (Latvian)
PS. Note also, that the adoption of surnames happened under the rule of Russian Empire, so it is likely that everywhere in the Russian Empire serfs were given surnames at the time of emancipation. HOWEVER, it appears that not everywhere in Russia the emancipation happened at the same time.