I’ve had my database open for the whole day, and when I looked in Finder, I saw the Date Modified was the last time I closed it. Is it autosaving at some regular interval? If my computer crashes, will I lose the data entered since the last time it opened?
Databases don’t autoSave. Data is being written inside the internal structure of the database as you work with it. When an application crashes (not just DEVONthink), there is always the possibility of losing some data. That’s just part of computing life and also why we still advocate manually saving files as you work on them. “Save, and save often.”, as the old adage goes. 
How does one manually save a file inside DTPO?
Command-S or Data > Save.
So, just to be clear: if I use Data/Save it saves all the changed files in the current database?
Data > Save All (⇧⌘S)
This saves only the files that are “dirty” (black circle inside the red dot in the stoplight display in the upper left corner of the document window means unsaved changes) and is applicable to documents you are editing internally in DEVONthink – such as RTF, Text, Formatted Text, HTML, Markdown, etc. It does not apply to documents opened from DEVONthink to an external editor (you have to save from that editor).