Will renaming a `content.php` file cause any problems?

There shouldnt be any issue, dont forget to change the slug in the template part call:

get_template_part( 'format', get_post_format() );

the best practice is to have them in their own folder:

get_template_part( 'template-parts/post/format', get_post_format() );

you can see the folder structure of twentyseventeen just change content respectively.

How get_template will work:

get_template_part will do a PHP require() for the first file that finds, it will create an array the first item array will use slug and name like this:

{$slug}-{$name}.php

the second item array will be:

{$slug}.php

the function will iterate the array of file names till it finds the one that exists.

so if you do a:

get_template_part( 'format', get_post_format() );

and for example the format is video the array of file names will be:

[0] "format-video.php"
[1] "format.php"

it will fallback to format.php, in the case of a Child theme the priority will be like this, with the video format as example:

[0] "format-video.php"  //in child theme
[1] "format-video.php"  //in parent theme
[2] "format.php" //in child theme
[3] "format.php" // in parent theme