Is it common between the European countries to have national bird officially stated?
France has no official national bird. The Gallic rooster has been used to represent France for a very long time but has no official status. From Wikipedia,
The Gallic rooster (...) is an unofficial national symbol of France (...). The rooster is also the symbol of the Wallonia region and the French Community of Belgium.
The official website of the Elysée palace, the official residence of the President of the French Republic while in office, has more information about the history of the rooster as a symbol of France. In particular,
Proposé comme emblème à Napoléon Ier par une commission de conseillers d'Etat, il fut refusé pour la raison suivante : " le coq n'a point de force, il ne peut être l'image d'un empire tel que la France ". (...) Naturellement dédaigné par Napoléon III, il devient un symbole quasi officiel sous la IIIème République : la grille du parc du Palais de l'Elysée construite à la fin du XIXème siècle est ornée d'un coq et la pièce d'or frappée en 1899 également.
Proposed as an emblem to Napoleon I by a commission of state councilors, it was refused for the following reason: "the rooster has no strength, it cannot be the image of an empire such as France". (...) Naturally disdained by Napoleon the III, it became a semi-official symbol under the Third Republic: the gate of the Palais de l'Elysée built at the end of the 19th century is decorated with a rooster and the 1899 gold piece too.
The second article of the fifth (and current) constitution of the French Republic details what are the symbols of France:
La langue de la République est le français. L'emblème national est le drapeau tricolore, bleu, blanc, rouge. L'hymne national est « La Marseillaise ». La devise de la République est « Liberté, Égalité, Fraternité ». Son principe est : gouvernement du peuple, par le peuple et pour le peuple.
The language of the Republic is French. The national emblem is the tricolor flag, blue, white and red. The national anthem is "La Marseillaise". The motto of the Republic is "Liberty, Equality, Fraternity". Its principle: government of the people, by the people, and for the people.
Is it true that there is no other European country without a national bird except Czechia and Slovakia?
The second question does not ask about an official national bird, so it is difficult to answer. What defines a national bird?
Wikipedia has a list of national birds by country, and which ones are official. Most entries have a source but they are difficult to evaluate (the German entry is linked to a website in German, the source for the gallic rooster is a seemingly unrelated Australian website,...).
Czech Republic and Slovakia are not in the list in Wikipedia, but neither are Portugal and Spain (thanks @SJuan for the comment).