Q: What was the constitutional status of Finland within Russian Empire?
Answer in one sentence: For the longest time Finland had a constitution (inherited from Swedish times/the Swedish constitution of 1772) while Russia itself had not 'a constitution', but the tsars kept respecting the Finnish one for this autonomous region within their empire.
The most basic characteristics of what happened to Finland within the Russian Empire during the 19th century is summarised on Wikipedia: Grand Duchy of Finland as:
During the Finnish War between Sweden and Russia, the four Estates of occupied Finland were assembled at the Diet of Porvoo on 29 March 1809 to pledge allegiance to Tsar Alexander I of Russia, who in return guaranteed that the area's laws and liberties, as well as religion, would be left unchanged. Following the Swedish defeat in the war and the signing of the Treaty of Fredrikshamn on 17 September 1809, Finland became a true autonomous Grand Duchy within the autocratic Russian Empire; but the usual balance of power between monarch and diet resting on taxation was not in place, since the Emperor could rely on the rest of his vast Empire. The title "Grand Duke of Finland" was added to the long list of titles of the Russian Tsar. […]
The Russian emperor ruled as the Grand Duke of Finland and was represented in Finland by the Governor-General. The Senate of Finland was the highest governing body of the Grand Duchy and was composed of native Finns. In St. Petersburg Finnish matters were represented by the Minister–Secretary of State for Finland. The Senate had a primarily advisory role until it got the right to representation in 1886. On top of having its own central, regional and local administration, Finland had its own stamps, currency and army.
Alexander I did not want the Grand Duchy to be a constitutional monarchy but the governmental institutions born during the Swedish rule offered him a more efficient form of government than the absolute monarchy in Russia. This evolved into a high level of autonomy by the end of the 19th century.
Notably, the entirety of exact details of this 'autonomy' within the Russian Empire were never put into detailed written laws. In effect, 'the constitution' of Finland was therefore the continuation of the Swedish "Instrument of Government (1772)" on the land of the Finnish Grand Duchy within the Russian Empire.
Reading the Swedish constitution, we see a lot of 'the king of Sweden shall vs the people have the right' etc. Like for example $1 proscribing that the king's and the country's religion shall be protestant, or $5 reading:
The king may govern and rule, save and protect the city and the country and all his and the crown's right, as law and our government form says.
Of course 'the king' in the first instance is the Swedish king, who may well remain protestant, for as much as any tsar would care, but in the second example, it is of course now the tsar who is the nominal ruler over Finland now…
For the new status this means that in principle, all occasions of 'Swedish king' shall be read as replaced with 'the Tsar of all Russia'. With the tsar simply ading the title 'Grand Duke of Finland' to his list of titles, of course…
This was of course only the beginning and 'naturally' tensions arose. These resulted from the one side to rising nationalism of the Fins seeking ever more autonomy and even eventual independence. Promised vision and material reality were of course not really congruent in the beginning, perhaps best exemplified by the Finnish diet not assembling until 1863, but the local executive was in any case the Senate of Finland…
And on the other side an official imperial policy was enacted that aimed at the very opposite, the
policy of Russification of Finland (Finnish: sortokaudet / sortovuodet, lit. 'times/years of oppression'; Russian: Русификация Финляндии, romanized: Rusyfikatsiya Finlyandii) was a governmental policy of the Russian Empire aimed at limiting the special status of the Grand Duchy of Finland and possibly the termination of its political autonomy and cultural uniqueness in 1899–1905 and in 1908–1917. It was a part of a larger policy of Russification pursued by late 19th–early 20th century Russian governments which tried to abolish cultural and administrative autonomy of non-Russian minorities within the empire.
Russia itself did not have a constitution under Alexander I or Nicholas I (one constitution was planned and signed only in 1881, but thanks to a little assassination and a reactionary successor not enacted), but during this time, Poland under Russian rule had a constitution. One that was respected when convenient for the tsar, which was once, when the opportunity presented itself to Nicky I to be crowned King of Poland, but never thereafter. The Polish constitution consequently completely abolished after the uprising in 1863, with Poland coming under direct Russian rule. The 'Finnish constitution' under Russian rule was the continuation of the previous Swedish constitution, and fared a little bit better, but under constant threat and pressure.
We therefore see in effect different meanings of 'constitution' at play: a written set of basic laws actually named "constitution", with two examples of them for subunits of the Russian Tsardom (Poland and Finland), and one meant to simply embody the very practical opposite of the narrower political concept: 'the basic compositional structure' of an absolute monarchy like Russia itself (if we analyse 'Finland's constitution' with that of 'Russia's constitution' in the 19th century).
Meanwhile great efforts had been made to win over the people of Finland. A deputation of Finnish representatives visited St. Petersburg in the autumn of 1808. Alexander declared himself determined to uphold all the existing liberties of the Finnish estates and people. He summoned a Finnish elected Diet in Borgå, and came personally to address it on 27 March 1809. From this time Finland was united with Russia only through the person of the monarch: Alexander was emperor of Russia and grand duke of Finland. Finland kept its own laws and institutions, and for nearly a century the Tsars respected them.
— Hugh Seton-Watson: "The Russian Empire 1801–1917", Oxford History of Modern Europe, Oxford University Press: Oxford, New York, 1988. (p114)
The details of most key documents for the above compiled in English language in the following, preceded with:
In uniting Finland to the Empire, Alexander I, anxious to secure a strategically vulnerable frontier area, chose to adopt a policy of consultation and conciliation. Whilst the war against Sweden still continued, representatives of the four Finnish Estates were summoned to a Diet in the small town of Porvoo. Here the Emperor promised to maintain and uphold the rights and privileges of each Estate in accordance with the 'constitutions' of the land. In return he expressed the hope that his new subjects, elevated to the rank of nationhood, would respond in a loyal and obedient manner to their new sovereign ruler.
Alexander's use of such politically loaded terms as 'constitution' and 'nation' was to provide posterity with ample material for debate. Finnish jurists and historians saw the Emperor as clearly having recognised the existence of a body of fundamental laws, which became the Finnish constitution. Russian nationalists countered this by arguing that the Autocrat would have been denying the very basis of his autocracy by turning himself into a constitutional, limited ruler in one part of the indivisible Russian Empire. In their eyes Alexander I had merely sworn to observe the existing rights and privileges of the Finnish Estates: he had not agreed to a wholesale adoption of the Swedish constitution. Both sides examined the semantics of the 1809 settlement in lengthy polemics, tinged with the nationalist and constitutionalist views of an age long exposed to these twins of the French revolution. But it is highly doubtful whether these ideas had begun to settle in the Finland of 1809. In essence the relationship sealed between Finland and Russia by the oath of loyalty of the Estates and the Imperial charter or gramota, was one of absolutist paternalism recognising the corporate rights of the Finnish Estate. The representatives of these Estates met as members of the community, anxious to preserve their own corporate interests. Raised in the traditions of Swedish aristocratic constitutionalism, their world view was monarchic-patriotic, and the suggestion that they were gathered as representatives of the sovereign Finnish nation would have been utterly alien to them.
— D. G. Kirby: "Finland and Russia 1808–1920. From Autonomy to Independence. A Selection of Documents", Macmillan: London, Basingstoke, 1975.