Hi Brad
Firstly it is great from a usability point of view to have a custom 404 page and I would link it to your most popular content and maybe add a search feature on the page for your site to help find the content that is missing. I have come across some nice 404s that actually have very concise sitemap in order to help the visitor navigate the site.In order to prevent Google from indexing your 404 page you need to make sure it returns an actuall 404 HTTP status code.
In order to understand how Goolgebot crawls your site I would look at the following post from Google themselves - https://support.google.com/webmasters/answer/182072?hl=en
Rather than being concerned about a 404 page having links on to keep the crawl going make sure you have an XML sitemap that you have submitted to Google via Webmaster Tools as this will help your crawl process.
Googlebot alots a set amount of time to crawling your site and it doesn't just stop crawling because it encounters a 404 error. However make sure that you monitor Google Webmaster Tools and take care of any reported 404s with 301 redirects for instance if the page has changed location. You will notice that Googlebot reports 404 erros on the days it finds them and these can often be multiple 404 errors encountered in one visit to your site by Googlebot. Keeing an eye on this and making sure you keep it updated will make your site as crawl efficient as possible which is clearly what you are after - as we all are
I thought this would also be interesting reading in relation to this - http://googlewebmastercentral.blogspot.co.uk/2011/05/do-404s-hurt-my-site.html
Hope this helps