As long as there have been no changes to the live since since you took the backup of it to work on in your dev environment, then yes you can replace the existing database with yours.
You do not need to activate the new theme on the live server – the currently active theme is stored in the database, which you will be overwriting with your database backup from dev.
You will need to make sure the database name, database username and database user password are changed if you were not using strong/safe passwords/names in dev.
You should install any plugins on live that you might have added to the site in dev before replacing the database.
You will also need to place the theme & child theme files on live before replacing the live database.
While working in your dev environment are you using a hosts file to get your urls to resolve correctly? If not and you did a find & replace in the dev db to be able to effectively use urls like http://localhost/category/this-is-the-url-of-my-post then you will need to reverse this and do a find & replace on the dev db for http://localhost and replace it with the live url.