What relation did Cossacks have to the Muscovite and Imperial Russian state?

What relation did Cossacks have to the Muscovite and Imperial Russian state?

First, who the Cossacks were exactly and their genesis is still hotly debated topic and that influences the answer, as Cossacks of latter days are not deemed "true cossacks" by some historians. Another problem that Cossacks weren't that homogenous and there were several factions with quite different views - see below.

Relation has changed over time - a lot. At first, Cossacks were completely independent. All "relation" they had was some annoyance to russian landlords (as they harbor russian criminal fugitives and escaped serfs). Annoyance was tolerated in exchange for some border protection. Still, Cossacks were unruly and were raiding russian proper as eagerly as non-russian.

This "general unruliness" caused a lot of headache to Russia once it get to imperial stage and started to have diplomatic relations with Persia and Turkey. Quite often, russian diplomats would manage to sign a peace accord with, say, Persian Shah only to have it quickly denounced by angry persians - after Cossacks raid to persian territory or attack to merchant caravan. Persian expedition of Sten'ka Razin - for example.

So, over time both Persia and Turkey started to insist that Russia have to take an obligation to reign in Cossacks freedom (to plunder caravans at least). On several occasions Cossacks raiding party got refuge on russian territory (defined at that time by line of military forts - остроги) while being pursued by shah's troops.

Cossacks of that era have mixed reputations in Russia - on one hand, they were seen as free, well-heeled and respected warriors who were often saving "christian" peasants from turkish slavery. On another hand - as unreliable allies, greedy and treacherous.

For example, then-General-Esaul Ivan Mazepa accused then-Hetman Ivan Samoylovych in treason and preparations to secede from Russia. Samoylovych was sentenced to death and Mazepa becomes Hetman. He successfully enlarged Cossack Hetmanate across Dnieper and that provided him support of Peter The Great.

Another story - how Vasyl Kochubey was put to death after his young daughter Maria became a lover to 63-years old Mazepa - Kochubey took affair as bloody insult and incest, since Mazepa was the godfather to Maria.

Then Sweden king Charles XII invaded Ukraine. Kochubey repeatedly warned russian czar Peter The Great about upcoming treason from Mazepa, but Peter didn't believe him. Meanwhile, Kochubey correspondence with czar was intercepted by Mazepa and Kochubey was put to death after torture. Mazepa rather quickly turned his coat to swedes in 1708, after that Kochubey was exonerated.

Only ~3000 cossacks followed Mazepa to Charles, while loyal to russian czar cossacks elected new Hetman - Ivan Skoropadsky and fought for Russia (their numbers were much higher, ~15000). Mazepa fought along swedish army against russian army at Poltava, lost and fled to Bendery with Charles (city was held by Turkey at a time). He died soon thereafter.

Peter The Great was quite angry with Mazepa's treachery and he razed Sich. Cossacks paid dearly for Mazepa's decision - Hetmanate was abolished and later "registered" Cossacks were moved to Novorussia (region between Caspian and Black Sea at west-east, Don river and Caucasus mountains at north-south).

This move started another period - of growing allegiance to Russian Crown. Cossacks received large land plots and enjoyed great deal of self-governance in exchange for military service. Cossacks participated in all operations of Russian Imperial Army. Gradually they became most loyal and therefore could be relied upon in sticky situations - like suppressing demonstrations against czarism. Also, new Cossacks regions popped up along russian frontier - Siberian, Seven-Rivers, Amur, Terek.

Don ataman Krasnov (ataman - that's how hetman sounded by 20th century) wrote a letter to Kaiser Wilhelm offering help and asking for support in 1917. Actively collaborated with Kaiser forces when they occupied Ukraine in 1918 and came into contact with Don Cossacks. This caused quite a friction with Alekseev, Denikin and Entante reps. Krasnov later wrote another letter - to Entante, after Central Powers were defeated. So this trait of maneuvering between powers seems to be persistent in Cossack leadership.

Did they see themselves as subordinate to the Tsar, or more like equals in some sort of partnership?

(Most of times most of them, since 1709 - caveat) saw themselves as in partnership, but not exactly on equal footing to Russian Emperor. Cossacks thought of themselves as a separate nation, their relation with russian, ukranian peasantry and jews was complicated - to put it mildly. They tended to be xenophobic and to dislike all non-Cossacks, calling them иногородние ~ outsiders. Whereas earlier, in 17-18 centuries it was easy to become a Cossack just by getting into Cossack-held region and "proving himself", by 20th century it was all but impossible, as a flood of newcomers from main Russia overwhelmed Cossacks.

Sources

History of Russia From Earliest Times, By Sergei M. Soloviev. Russian Imperial historian is certainly biased toward Russia and Romanovs, but traces all major events.
Russian Civil War, By Evan Mawdsley. Krasnov's affair with letters.