Database updates Out of date I downloaded a couple of module updates. Before I did so, I didn't disable the modules. Is this ok? I am now seeing the following in my Status Report:

Database updates    Out of date
Some modules have database schema updates to install. You should run the database update script immediately.

Is it safe for me to run the database update script now? Or should I first do a core update (am currently running 7.18 and would be upgrading to 7.19)?
Yes.  Go ahead and run update.php.  It will run the updates (which are usually but not necessarily schema updates) for the new modules.

There is no need to disable modules before updating them.  There is no need to update Core first.  You should, however, put the site into maintenance mode before updating the modules until after you run update.php.

You should run update.php every single time you update a module or the Core.