Updating to PHP 5.4 breaks Drupal 7.25 with Strict warning: Declaration on bootstrap.inc
These aren't critical errors right? 


No, it's just a warning. Not fatal. It comes from the E_STRICT error constant, which as of PHP5.4 was included in E_ALL. It's quite common to see those errors after upgrading 5.3 to 5.4 (not just for Drupal), as most people copy the bulk of their config from the previous version. 

The constant itself provides this to error reporting:


  Enable to have PHP suggest changes to your code which will ensure the best interoperability and forward compatibility of your code.





  What do the following errors mean exactly?


In Drupal terms, it means one or more contrib modules you have installed contain code that targets older version of PHP (or at least hasn't been updated recently). Specifically, PHP is complaining that the arguments or access modification of a method on a child class doesn't match the same method on the parent class.


  Can I just upgrade my Drupal 7 to the latest 7.31 as of 2014-09-24 to fix these issues?


In your case the warning is being caused by the messaging_mail module, so updating core won't do anything (though you should update ASAP, you're a long way behind). You could disable strict errors if you just want to hide the warning, but the best way forward is to report this as a bug to the appropriate module.