I see a lot of MySQL errors and a few PHP errors. Some of those have already been posted.
1. MySQL: Could you check if you are using "localhost" within your config-wp.php as the server address? Maybe the MySQL server is declining your connections at some point as you are using some other value, like a regular domain-name. So the database server treats your connections like external connections and stops them... might be a little far fetched, but worth a try.
2. PHP: There seem to be some Fatal errors regarding the amount of allowed memory for script execution. You could increase the memory value in .htaccess from what it is now to a value a little big higher. For example:
php_value memory_limit 512M
3. Downtime:
How long does your downtime take whenever the crash occurs? Just a few seconds or a couple of minutes/hours?