Can I use the same $con = mysqli_connect or should I create multiple connections?
Yes, usually you would just reuse it if it's to the same DB, same schema, same connection parameters.
When is it proper to use multiple $con = mysqli_connect?
When you have to connect to different databases, or when you need to connect using different parameters. Other than that, practically never. In the event that you need to run multiple queries, you could actually do this with one connection and mysqli::multi_query. See http://www.php.net/manual/en/mysqli.multi-query.php
What are the consequences of not "free"ing up the query before executing the next SQL?
It's a best practice to close your result as soon as you no longer need it. That said, I know people don't close results or connections explicitly, but rather they let the connection get reaped when the PHP page finishes processing automatically. This can burn you pretty bad in Java when pooling connections. Not so bad in PHP, especially when not pooling. https://www.php.net//manual/en/mysqli.query.php
How about do an experiment. Don't do this on a production database, but find a Dev/sandbox database and write some PHP page that opens 200+ connections without closing them in between. Watch what happens on your PHP (i.e. Apache) server, and also watch what happens on your DB server when this happens. Repeat with explicit closing between each connection opening.
If I was not, I assume I could just use databaseschemaname.tablename in my SQL?
Yes, as long as the db user you are connecting with has privileges on the other database or table, you can do this.