Can open-source software be peer-reviewed and published?

Yes, Open Source software can be published. What's required varies depending on the venue.

There are general journals that focus on the software process. The idea is to encourage better software development gets the credit it deserves. Examples of journals with this approach are the Journal of Open Research Software and the Journal of Open Source Software.

Then there are domain specific journals that have specific software paper policies. An example would be the AAS Journals which state

AAS Journals welcome articles which describe the design and function of software of relevance to research in astronomy and astrophysics. Such articles should contain a description of the software, its novel features and its intended use. Such articles need not include research results produced using the software, although including examples of applications can be helpful.

(Emphasis mine to link to a point in the original question)

Yes. Software can be published as an open source tool with a peer review process. Several tradition-styled academic journals exist. Given the OP's profile, here are some journals that publish R packages, genetics tools, or environmental software:

• The R Journal
• Journal of Statistical Software
• Environmental Modelling and Software
• Molecular Ecology Resources
• Source Code for Biology and Medicine.

Also, some government agencies have formal peer review processes for software and code for public release (e.g., the US Geological Survey, the US Department of Energy).

Sure it's publishable. You write up a paper detailing what is in the code, how to use it, examples, potential problems, and so on.

Here's an example, and here's the Github link to the source code.