Can I plug in an iPhone 5s with no screen? I have an iPhone 5s I am repairing,

I want to back it up,

but it has no screen.

Or home button.

I know how to get the pictures off, but can I boot up the iPhone and plug it into my iMac even if it has no screen?
Yes, you can. As long as the iPhone itself you know is working, you can plug it in. 

The iPhone is now listed in Image Capture, even though I now have to find out how to unlock the phone...

Either way, never fear, the iPhone will respond even if it has missing parts :-)