Hello Sean,
I am here to help you sleep at night. 
For your search feature I suggest following this guide .
Hooking it up to your products collection should not be a problem, just change the query to . query(‘Store/Products’) , or however your products are structured.
I have no idea how to do that without having a blank ugly list on the main page before anything else, and I don’t think you could do that. Could you?
Yes this is possible, just let the default value be all products, so you avoid the blank ugly list.
What do you guys think is my best option here? Can I make a search bar in the header and always display results on any page? Will a search bar run with a list/grid and not database?
You’re best option is to use the search feature found above and put it on specific pages that you want, only if it is imperative, keep it in the header as it is a big resource to have on all pages. The search bar will query the actual database and update the list/grid based off the filter result options.
Try it out, and if you run into any trouble let us know, we will help you debug it!
Best,
Majd