Would there be copyright issues?

I was worried about this but from my research using book covers is acceptable if it's for the purpose of review. You can't use any other material or graphics form the book.

would these be scanned in by the user?

Possible for the very first version this could be sourced online but in future, I would like to implement book covers automatically based on the book title/version. I need to look into what options are available for this.

How would it work with old out-of-print books, eg, books read via online libraries like Project Gutenberg, or simply downloaded as pdfs?

I had not thought of this. Perhaps there are already images out there for these kinds of resources that we could match content with.

General image for illustration if there was no actual book cover available?
would need to be careful with copyright restrictions here but as long as the uploader has rights to the issue I don't see this as a problem when there is no other cover available.

I would also place importance on the searchability, ensuring that books can be found even if you don't have every single word or apostrophe.

I agree. Search is important and a great feature. It probably won't be added for awhile but the aim is to have it in future.

I'd like a mechanism whereby review authors can create genre categories, maybe similar to the Steemit tag system.

Totally possible.

thanks for all of your ideas and feedback