SQLite is an embedded database; it does not run as a separate server process, but is a library that is usually compiled directly into the application itself.

Running the sqlite3 command-line shell does not make sense, because the version of the SQLite library compiled into that is not necessarily related to the version compiled into the Ruby gem. The guide you linked to is wrong; just ignore that point. (But if you want to test SQL queries outside of Ruby, there's nothing wrong with installing and running sqlite3.exe.)

You do not need to install the DLL manually; any program that needed it would already ship with it.

For how to test the SQLite Ruby gem, see the sqlite3-ruby documentation.

You can install SQLite on Windows so that you can use it on terminal(command prompt).

First, download the 3rd one saying A bundle of command-line ... from the website:

Then, unzip it, then get 3 files below, then put them in C:\sqlite\ which you need to create manually:

sqldiff.exe
sqlite3.exe
sqlite3_analyzer.exe

Then, search and click on Edit the system environment variables:

Then, click on Environment Variables...:

Finally, set C:\sqlite to Path in both or either User variables and/or System variables, then you can use SQLite on terminal(command prompt). *User variables are specific only to the currently logged-in user and System variables are globally accessed by all users: