Sharepoint - Sharepoint Online default search result/scope in the small "global" search box

To achieve the goal, you can go to configure search navigation.

Here are the steps for your reference:

1. Go to the Site settings page, click Search Settings under Search.

2. In the section of Configure Search Navigation, click Add Link to add the the following links:

   1). Title = Everything; URL = /search/pages/results.aspx

   2). Title = People; URL = /search/pages/peopleresults.aspx

   3). Title = Conversations; URL = /search/pages/conversationresults.aspx

   4). Title = This Site; URL =/_layouts/15/osssearchresults.aspx Note: The top option will be the default.

3. Click OK.

Note: It's available for the site level, not the site collection level. That is, if you want to set this requirement on all sites in the site collection, it's required to set them one by one.

http://community.office365.com/en-us/f/154/t/244788.aspx

http://social.technet.microsoft.com/Forums/sharepoint/en-US/0d0f69f8-b461-4bd1-896e-3ef8c06b8036/set-everything-as-a-default-scope-sharepoint-2013

Yes you can. Go to Site settings > Site collection Administration > Search Settings on the site where you want to be able to search for everything. ( You have to do the following steps for every sitecollection. )

Set "Search Center URL" to your global search center url. ( url/sites/searchcenter/pages)

Uncheck "Use the same results page settings as my parent" and then click the radio button "Send queries to a custom results page url." ( url/sites/searchcenter/pages/results.aspx )