Hi there, so I’ve had this question saved for awhile and got side-tracked by a number of personal issues, but I’m here now!

The reason why this is a difficult question to answer is because the Venetian system of government is evolved continuously over the years and centuries, with frequent ad-hoc exceptions or modifications. A multitude of evolving customary practices and no single formal written constitution complete the picture, meaning that any answer to “how something-or-other functioned” is either a snapshot of an arbitrary moment in time, or a long and laborious survey. The context in which these institutions existed additionally cannot be ignored: More so than procedure and formality, the government institutions in venice were built on trust (or lack thereof), social and familial ties, and responses to real or perceived threats; each of these factors guided and impacted how these institutions functioned. What this ultimately means is that depending on the specific moment in time a given source is referring to, you might very well get different answers to the same question.

So it goes without saying, the Council of Ten, like other venetian institutions, changed over the years: in the wake of Bajamonte Tiepolo’s revolt (the only internal revolt in Venice’s history) while the populace had remained indifferent to the attempt at usurpation, the Venetian ruling class was nonetheless spooked into establishing the body to counter subversive or otherwise rebellious actors in the city. It’s also another step in a trend of increasingly oligarchic government in Venice: the idea in the venetian establishment that certain kinds of people ought to be entrusted with the business of government, while other kinds of people should be kept out. There was nothing particularly more oligarchic about about the Council of Ten than the Major Council (the body which every eligible venetian aristocrat had right to sit in) which had already been “Sealed” in its membership, but Venice’s political history had until then focused on inclusive consensus-building institution (hence the love of “Councils,” however specialized they might have been) and a directly elected council with broad (“Non-Specialized”) powers was a new development.

The powers granted to the Ten are so broad that although initially envisioned as a sort of tribunal or investigative body, they quickly got involved with a whole number of executive-level activities, to the point that they effectively replaced the Doge as the Republic’s executive. Some popular historians like to use the analogy of separation of the political Executive with the Head of State, but this simplistic if convenient analogy is only really valid a few centuries after the Ten were established (and the institution had more firmly established its authority). The analogy which might work best for the early Ten is more akin to a powerful cops of auditors, taking an interest in the various councils of the Senate or the actions of the Doge, and issuing advice (or orders) as they felt was necessary.

But the Ten weren’t all-powerful. They had oversight over Senate committees, but they were accountable to the Major Council (from which the Senate was drawn). The Doge could, in theory, appeal to the Senate or Major Council should he be at odds with the Ten. And the Ten were indeed sometimes reprimanded, or their purview was explicitly realigned. Notably, over the course of the 16th century (but also at moments in the 17th and 18th centuries) the Major Council passed decrees to more formally define (and ultimately to curb) the power of the Council of Ten. Nowadays, the verdict on how much power the Council of Ten functionally yielded might be described as instilled on a pendulum: Sometimes they seemed all-powerful, other times they seemed to be an ordinary organ of the Senate. The actual answer, as you may expect from the discussion above, depends on a multitude of varied factors (it's most likely that in some moments the Ten was in a position of strength, in other moments it was eclipsed or even curtailed by other organs of government).

So to answer your specific questions:

Was it one of the major executive bodies? Sort of. The Venetians never really reasoned in terms of modern separation of powers as a contemporary government would have, and thus while the Ten did perform executive actions, they also performed advisory actions, investigative actions, and frequently passed judicial decisions.

Were elections annual? As far as I know, yes. However, perhaps in a moment in time they were semiannual (I really don’t know). Again, the Venetians had few formalized protocols and no constitution. If at a moment in time they felt the need to elect councilors semiannual, they very well might have done so.

Were two members of the same dynasty were forbidden from sitting on the council at a given time? Probably, although I haven’t read anything to this effect and don’t know what your source is. All venetian political dynasties were interrelated to some degree so it’s more likely that the reasoned along the lines of economic and social groups, rather than exclusively by surname.

Did the council have three rotating chiefs which were not allowed to leave the Ducal Palace? Again, it depends on the moment in time. At one phase they might have, but I also had read that they rotated weekly. It’s possible that the rotation was initially weekly, but at some point transitioned to being monthly.