DownUp

No. It only rewrites one line (the Version: header) in the target plugin's or theme's main file. The actual download and install still happens through WordPress's own "Update Now," using either WordPress.org or the target's own legitimate update source — never this plugin.

It appends a -dev suffix to the current version (e.g. 1.4.2 becomes 1.4.2-dev). Since that string compares as lower than the real version, WordPress's normal update check reports an update is available, the same way it would for a genuine new release.

Decrementing a number (e.g. 1.4.2 to 1.4.1) risks colliding with a real version that was actually released before. Appending -dev is always guaranteed to compare as lower without guessing at arithmetic or matching a real past release.

Nothing happens the second time — the plugin detects the version already carries the -dev marker and refuses, asking you to finish or cancel the pending update first.

"Prepare Update" only makes your installed version look like a pre-release of itself. WordPress still needs an update source that knows about the real version, so an update only shows if the latest version that source reports is equal to or higher than the version you had installed. The most common cause is publishing a brand-new release and preparing it straight away: WordPress.org's update check can lag behind the plugin's own page for a while, so it still reports the previous release — which is lower than yours, so there is nothing to offer yet. Wait a while, then go to Dashboard > Updates and click "Check again". Also note that a plugin or theme with no update source at all (not on WordPress.org and without its own updater) will never show an update, however it's prepared.

Go to Tools > DownUp and uncheck it in the Plugins or Themes list there. Everything is enabled by default; unchecking an item just hides the "Prepare Update" link for it.

Both, but not identically in the UI. Plugins get a "Prepare Update" row action right on the Plugins page, because WordPress supports custom row actions there. Themes don't have an equivalent — WordPress's Themes screen is built from JS templates that only recognize a few fixed buttons (Activate, Customize, Delete), so there's no way to inject a custom action into it. Themes instead get their "Prepare Update" link on the Tools > DownUp screen itself. Either way, the underlying mechanism is identical — a -dev suffix appended to the Version: line — and the resulting "update available" notice still shows up normally wherever WordPress would normally show it (the Plugins list, or the theme's tile on Appearance > Themes).

It only touches the item's own version header, nothing else, and requires the update_plugins (or update_themes) capability plus a nonce on every action. That said, it will happily "prepare" anything you point it at — including things you didn't mean to — so it's worth keeping the settings checklist limited to what you actually manage this way.