Website keeps going down, is it a plugin, database, memory problem, or something else?

Allowed memory exhausted is generally generated because the server can’t handle the memory usage of your site, and the actual memory issue can, as Pieter Goosen mentioned, really be anything on the site.

Answers to your questions:

  1. It’s not too many plugins. WordPress can handle it 🙂
  2. Probably not. If the storage of your website is very very limited this would be a problem, e.g. if you have a database of 10GB and your website has a storage of 10.2GB, then of course this would be an issue.
  3. It’s obviously the W3 Total Cache plugin that is generating the error, but that doesn’t mean it’s a problem in that plugin.

I would recommend these steps to debug…

  1. Check with the host if it’s possible to change the memory limit on the server. ini_set('memory_limit', '-1'); would set memory for unlimited usage, but if it’s a shared host this would probably not be possible.

  2. Copy your production site (with a plugin like Duplicator) to a local server. Is it
    the same problem with it being slow?

Do the testing on your local site…

  1. Try to deactivate the plugin (as you usually deactivate a plugin). Check in a browser, being sure to refresh the cache! Alternatively, if you have the option, check in a totally new browser that you usually do not use (just to be sure that it’s not the cache that is messing things up).

  2. Try to remove plugins one by one… and test.

  3. Remove the W3 Total Cache plugin. This is a good tutorial on how to achieve that.

These above steps I think/hope will give you a clue what’s going on.