SmallSearchInputBox does not seem to show the same query suggestions(global query suggestions manually added through central admin are shown) as the search box in the search center. Furthermore, it shows different suggestions at the root site and at the sub-sites. Users end up seeing different query suggestions at different levels of the site hierarchy. I knew query suggestions were aggregated at result source and site-collections levels. But this behavior seems to suggest that query suggestions are stored at individual site level.

Also tried to use the SearchBoxScriptWebPart instead of the delegate control and different settings but that did not help either.

Small search box work differently at different level. When used in list it will search in list, when used in subsite it will search at subsite level

Search center always search across site collections at same time. 

What you described is what translates to contextual search scopes from 2010. But you can also set the search center and custom results page at the site level. With this set, the search box uses the search center to handle your queries. However, my concern was not about queries, but query suggestions.

So when you change the search results setting it does not mean you change the query suggestions for that search box. 

Search results and Query suggestions are 2 different settings.