I'll answer them one at a time.

Has skepticism towards scientific discoveries always been present in societies, or has it changed over time?

One could argue that popular opinion towards scientific discoveries has shifted away from suspicion towards acceptance. Upto early modern times, things which could not be understood were often ascribed to the supernatural. There are plenty of accounts of people who were scared of the first trains, cars or even radio's. Because they did not understand how these things functioned, they would call them "demonic" or "magic". Scientific principles weren't at the forefront of their mind or education, as they are in our time. With mechanical objects permeating every aspect of life, the rise of educational levels, the improvement of long-range communication methods and finally the popular acceptance of scientific princples; mechanical objects or abstract concepts which could not be understood were now simply explained with the concept of "science".

Has any work been done on investigating how widespread skepticism has been over different periods of time?

There are works discussing suspicion towards specific inventions or discoveries, but you will be hard-pressed to find research discussing the level of scepticism throughout history. Why? It's considered to be bad historical practice or atleast too ambitious to research something that subjective and broad. You can't objectively assess the level of scepticism while still doing justice to the very specific circumstances of individual cases. That being said, you might be able to find some comparative research. For example a paper which compares multiple research endeavours with the reception of new inventions as their subject. That being said, the first part of my comment is mostly influenced by articles discussing the acceptance of scientific discoveries. So it's a fairly popular subject in general accounts of early modern or recent history.

Also has it been restricted to a few topics, or have we simply forgotten about skepticism towards discoveries that now seem unchallenged?

This is a very interesting question which more closely relates to my expertise. There are some historians who focus on the theoretical side of history who have pointed towards the existence of "black boxes" in our collective minds and even within academic practice. When a theory first arises, it's still being criticized and challenged. After some time, many theories become more universally accepted and might end up in a "black box". They become part of the foundations of our thinking and won't -or even can't- be challenged or criticized anymore. It's a lot more complex than this, but that's the jist of it. It's a fairly difficult matter, so elaborating might just be counter-intuitive. That's one reason why history is so important and why it's often a matter of principle for historians to keep questioning their own fundamentals.