How to tune a stored procedure that is fast on test data but slow in production

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SQL-Server

If you have a database (SQL server 2012) with testdata in development and preproduction phase and the real from the company's server take place only in the production phase. In other words, the real data take place only the the production phase and not in the test and pre production phase.

Production phase take place inside in the company's server.

When you have a stored procedure that takes a lot of time to retrieve the data in the production phase and you need to make a performance tuning.

This specific stored procedure is part of critical function for a webpage. The SP is used daily.

My questions are:

1. Is it recommended to do a performance tuning in the production phase in relation to real data?

2. Is it something that I should be aware of? Today, I'm a newbie in performance tuning.

3. What plugin or tool do you use in order to review the the database and its specific stored procedure's performance?

Thanks!