1 How common is this no-article use? Can you say either "Dog is a domesticated species" or "The dog is a domesticated species" whichever you fancy at a given moment?
No, we can not.
Pigeon is the most widespread species of bird in European cities can be paraphrased as
The most widespread species of bird in European cities is called pigeon (not a pigeon, and not the pigeon).
The bare singular pigeon merely names the type of bird. The sentence does not generalize over pigeons, does not characterize them in any way and, as a result, it is not a generic sentences. That’s why bare countable singulars are not used taxonomically. (Exceptions? I can’t think of any.)
But bare countable singulars are used often to names types: we eat breakfast (not a/the breakfast) and do things after breakfast every day (names one type of meal), play oboe (names a type of instrument), go to jail (names a type of detention), hunters hunt bear (names a type of game).
2 You'd better mind your spelling before zielonosiwy pidgeons you to death.
The zielonosiwy type is called dumbbell. Saying
he is a dumbbell would be wrong, because
a dumbbell implies possible particularization of
type dumbbell. The article a is an ostensive sign of more than one dumbbell in existence; the type name (no article) dumbbell subsumes every conceivable thing that is called dumbbell and rejects particularization. It is in reality a proper name and should be capitalized: Dumbbell.