Dropping the connection is indicative of an overloaded shared server, as well as "Whilst the site is speedier when no plugins are included, we go back to the same issue when I add plugins back in one by one." You can't really do anything about that.
Running out of php memory can possibly be fixed, depending on how much access GD gives you: https://codex.wordpress.org/Editing_wp-config.php#Increasing_memory_allocated_to_PHP
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