BEST PRACTICES: updating a theme on a live site without bringing it down

What are the best reliable practices in doing this?

I exported the SQL from the live site and installed it on my local drive in MAMP.
I installed it and forgot to change the WordPress URL and Site URL fields. Disaster.

The live site somehow was referencing my new theme I was testing but because it was pointing to localhost, couldn’t produce a stylesheet so the live site was looking like the .css file went missing.

After noticing this, I lost ability to even login the live site’s wp-login.php file.

Fortunately I found some tips to add some lines in the functions.php of my theme to “reset” the live site.

I really was lost but recovered and restored it. I have since disabled my MAMP version till I figure this out here.

I want to know what I should do in order to run another site so I can show the customer what his site will “look” like with the new template?

As not all templates are built the same, I don’t want to even put it on a live site as I know some categories (posts vs. pages) may be all wrong and the template is destroyed from the demo of the purchased site.

What do you recommend in terms of showing a sample short of fiddling on the live site?

Also, When I tried to merge the SQL file to my MAMP, I had to fiddle with php.ini because the SQL file was 350MB when the php.ini settings to upload were only 32MB.

Anyone know what I can do to improve importing large SQL files also?

Thank you in advance!