Re: Free/Libre and Open Source advantage
"I made an Android app that reads the clipboard on startup without asking. It does it to see if the clipboard contents is a URL, in which case it pre-populates a text field with that URL"
That's your choice, but I think it's not that useful for a few reasons.
"I could have added an extra Paste button, but then I'd have to worry about issues like "would the button take up too much room on small displays" (or if it's hidden, would users be able to find it)"
This is why consistent UI decisions across apps are so useful. In most operating systems, there is a typical way to copy and paste and any app supporting those actions does it the same way. If your paste function was available and followed a convention, most users would know how to use it.
"and "would users be confused if pressing the button causes an error message because the clipboard contents is not something we can handle—would it be better to simply not offer the option in that case?"
As I see it, this is a nonissue. If the clipboard contents are not text, you just don't paste any contents into the box. If it is text, you put it in when they paste and let the user decide what to do then. After all, you already have to have some control for a user entering an invalid URL, either a typo or testing you, so if the clipboard contents are invalid, you just report them as invalid just like you would if I mistyped the URL as "https;\\".
These are subjective decisions, and everyone will probably have their own opinion on what is best. For what it's worth, mine and yours differ.