Magento Admin Problems / Slow Loading / Crashing

We have been having a lot of problems with our admin and website. We often get random errors which we do not understand. 1st Error ------ 500 Internal Server Error Sorry, something went wrong :( 2nd Error ------- SQLSTATE[HY000]: General error: 2006 MySQL server has gone away Trace: #0 /home/alphacate/web/[login to view URL](110): Zend_Db_Statement_Pdo->_execute(Array) #1 /home/alphacate/web/[login to view URL](291): Varien_Db_Statement_Pdo_Mysql->_execute(Array) #2 /home/alphacate/web/[login to view URL](479): Zend_Db_Statement->execute(Array) #3 /home/alphacate/web/[login to view URL](238): Zend_Db_Adapter_Abstract->query('SELECT `admin_r...', Array) #4 /home/alphacate/web/[login to view URL](428): Zend_Db_Adapter_Pdo_Abstract->query('SELECT `admin_r...', Array) #5 /home/alphacate/web/[login to view URL](734): Varien_Db_Adapter_Pdo_Mysql->query(Object(Varien_Db_Select), Array) #6 /home/alphacate/web/[login to view URL](69): Zend_Db_Adapter_Abstract->fetchAll(Object(Varien_Db_Select)) #7 /home/alphacate/web/[login to view URL](100): Mage_Admin_Model_Resource_Acl->loadAcl() #8 /home/alphacate/web/[login to view URL](69): Mage_Admin_Model_Session->login('admin', 'LdcMJfRW9xtG', Object(Mage_Core_Controller_Request_Http)) #9 /home/alphacate/web/[login to view URL](1338): Mage_Admin_Model_Observer->actionPreDispatchAdmin(Object(Varien_Event_Observer)) #10 /home/alphacate/web/[login to view URL](1317): Mage_Core_Model_App->_callObserverMethod(Object(Mage_Admin_Model_Observer), 'actionPreDispat...', Object(Varien_Event_Observer)) #11 /home/alphacate/web/[login to view URL](448): Mage_Core_Model_App->dispatchEvent('controller_acti...', Array) #12 /home/alphacate/web/[login to view URL](527): Mage::dispatchEvent('controller_acti...', Array) #13 /home/alphacate/web/[login to view URL](160): Mage_Core_Controller_Varien_Action->preDispatch() #14 /home/alphacate/web/[login to view URL](407): Mage_Adminhtml_Controller_Action->preDispatch() #15 /home/alphacate/web/[login to view URL](250): Mage_Core_Controller_Varien_Action->dispatch('index') #16 /home/alphacate/web/[login to view URL](172): Mage_Core_Controller_Varien_Router_Standard->match(Object(Mage_Core_Controller_Request_Http)) #17 /home/alphacate/web/[login to view URL](354): Mage_Core_Controller_Varien_Front->dispatch() #18 /home/alphacate/web/[login to view URL](684): Mage_Core_Model_App->run(Array) #19 /home/alphacate/web/[login to view URL](87): Mage::run('', 'store') #20 {main}