When I type something into our search box I get redirected to _layouts/15/osssearchresults.aspx. How do I brand this page? Can I apply our master page and page layouts to it?
You can create a new display template.The display template defines what we want our results to look like.
To check all available search display template
go to Site settings --> Master pages and page layouts. In the Master Page Gallery, click Display Templates --> Search.
Check these examples for more granular steps.
Yes. You should use the same master page everywhere which will make SharePoint look the same wherever you go. The user experience will be greatly enhanced. If you have a custom master page you should use it on the search centers/search result page as well. If you just want change the design of the search result and not alter the content of the search result it is enough to use CSS for this which you include in the master page itself. Display templates are great but only needed when you want to change the content/behaivour of the search result.