500 Errors on CiviCRM 5.1.2 with Varnish on WordPress
You can certainly try to set up rules to bypass the reverse proxy, but at the same time, I think it will be more cost-effective to change hosting.  Varnish problems aren't going to be the only issue you have when you have managed hosting that is specially tuned for WordPress hosting.

CiviCRM and WordPress work nicely together because they rely on the same LAMP stack and it's valuable to have your CRM and website tightly integrated.  However, from another perspective, a CRM and a website have opposite performance needs.  A website needs to serve the same few dozen pages to thousands of visitors.  A CRM needs to pull from among thousands of contacts' information for a handful of users.

Managed WordPress hosting (whether through Dreamhost, WP Engine, or others) attempts to maximize website performance by caching as much as possible so that the most commonly-visited pages can be delivered whole without querying the database.  This tilts the process completely towards the handful of content, lots of visitors use case for a website.

For a CRM, there's the issue you're facing where you have to tell the reverse proxy to skip various pages and widgets that are driven by CiviCRM.  But when you fix that, at least for certain pages you've defeated the performance tricks that you're paying your host to implement.

The entire business model of this kind of hosting is to rely on these tools to get maximum performance out of minimal hardware.  Without the caching, you're just left with cheap hosting.  That's not going to serve you well as you try to use CiviCRM.

Your better bet is to host with something a little more substantial.  What my colleagues and I have recommended for years is a virtual private server (VPS or cloud server) with at least 4GB of RAM and a dual-core processor.  You can get these for around $20 per month at Linode, AWS, DigitalOcean, and elsewhere.  They require setting up the whole LAMP stack, but that's something lots of us who support CiviCRM can do.

Once you've done this, you'll have better performance, a stack that is more broadly supported and used, and the ability to set up a practically unlimited number of development sites, extra domains, and so forth.