Dragging a Mac app to the trash, does that remove everything?

After you've binned it, have a look in System Settings > General > Login Items & Extensions. Leftover background items from deleted apps sometimes stay listed there and you can switch them off. If you want to hunt down the leftover files by hand, in Finder use Go > Go to Folder and type ~/Library, then search the folders for the app's name. Only delete things that clearly belong to that app.

Mostly, yes. The app itself goes, but it leaves small preference and cache files in your Library folder (Application Support, Caches, Preferences). For most apps that's a few MB and doesn't matter. The exception is anything that installed drivers or background helpers, antivirus, VPNs, Adobe stuff. Use their own uninstaller for those, otherwise bits keep running after the app is gone.