A long time ago the church and the state were inseparable. In our time, is the same true for the sciences and the state? Does the state encourage scientific (and thus technological) developments and does science influence the state to police the state in a scientific way? For sure many efforts are made by the state to impose the scientific ideology on its citizens to ensure that for future generations the ideology will not vanish into thin air. And what about the concept of the panopticon, in wich the connection is very tight and even gave us new sciences. By the way three of the very few realizations of a panopticon are made in Holland, the country where I live. All three are prisons (wich gave birth to criminology) and exact copies of one another.
Just for a little history, there were two ways in which the government controlled religion. The first was the Establishment of religion, or the direct subsidization, which occurs still in the U.K. with the Church of England. The second is through the prohibition of other, competing religions. Obviously, this is still an issue in places like China or Saudi Arabia, and is an open question in Germany with Scientology.
If the sciences, natural or otherwise (there is no such thing as an exact science), had a relationship with government like churches did in the past it would take one of these two forms. In order to answer the question, we should therefore look at funding, and at prohibition on research topics.
The best data exist for the U.S., so I will rely upon that, but anecdotal evidence suggests that similar patterns hold across the OECD. While the Federal government is a major source for funding, it is far from the most influential. Furthermore, there are multiple agencies funding different, and even sometimes opposing research. Non-government research funding from endowments, private universities, foundations, and corporations, to name a few. In democracies, there is no 'establishment' of a 'set science.'
This form of relationship was common in the Soviet Union, but has died out for the most part. China, and a few other authoritarian countries, maintain more direct control on what is allowed to be studied, but even China has abandoned the idea of "Marxist" science, for the most part.
Two possible exceptions to this bear mentioning. Speech codes exist at a number of universities in the U.S.. These restrictions on speech can be seen as a prohibition on certain kinds of research, and there is anecdotal evidence that faculty and researchers self-censor to avoid running afoul of the dominant ideology. However for the most part, these codes are local or cultural and, as far as I know, are never nationally imposed.
Many countries in Europe have laws against holocaust denial and some in the U.S. have advocated using anti-corruption and racketeering laws to prosecute Climate Change skepticism. Both of these would be examples of prohibitions on certain research. For the most part, the laws against holocaust denial are of such limited scope, however, that they do not affect most research agendas whatsoever—after all, there aren't many topics or research which touch on the factuality of the Holocaust, or which might be advanced by problematizing it. Given the broad reach of subjects related to climate change skepticism, including but not limited to policy, biology, chemistry and physics and many organizations which fund them, if any government were to actually start prosecuting 'denialists' using RICO laws, it would be a much more intrusive prohibition, and would be the most significant attempt to directly control the sciences by democratic governments to date.
For the most part, there is not an analogous relationship between the sciences and the state in the modern era, in democratic countries. There is enough funding and a free enough research environment that nearly any agenda has a chance to flourish. Even in non-democratic ones, the relationship is not particularly strong, and a lot of good research has come out of less free locations like China.
There are, however, movements that could signal a shift in that relationship that would change the answer. Whether or not that would change the relationship to be exactly like the ancient one between church and state remains to be seen.
Like anything else, the sciences, be they 'hard' or 'soft' sciences, physical or social, can be controlled by whoever has the means to control information and manner in which is received. This is primarily a matter of what are known as information systems, and may be either physically, philosophically or systematically coercive in nature. However, it is worth noting that it is not necessarily the democratically elected public representatives nor even their executive, judicial or administrative branches which necessarily hold power...
Anyone who has the means to create and/or disseminate information may act to sway the sciences even if one believes such to be based on wholly objective criteria. At the widest possible level one can even begin to influence the very paradigms on which science functions...
We can, for example, question the very idea that a 'double-blind' trial even exists, or that it is possible to isolate for specific conditions, or that such isolation is useful in the real world. On other levels we could use the media, and manipulate it in various ways, so scientists feel unconsciously drawn towards certain subject matter and are moved to explore the world and its properties in different ways. On another level still, if we find ourselves in a position of directly coercive power (say, your supervisor at university while you do a PhD, or your lecturer's boss, who decides if she gets paid or not...) then we can just say 'no, you can't look into this, look into that instead...'
So yes, government can affect the sciences in the ways you outlined, but so can most people in most situations, where they have the will to do so. The degree to which they are able to, however, is a complicated thing to chart. Though the Katz centrality may give you some ideas in this regard.
Many consider the 'social sciences' to be little more than political indoctrination or expoundiation of ideology. For example, in university it is not unique to have social science professors directly support one form of ideology while directly attacking another. The 'hard sciences' ( which require verifiable testing) are immune from ideology.