Can you restrict core search to the current language? I can't figure out how to restrict results to current language other than crude theme preprocess hacks to inspect the language of individual results. I haven't had luck getting contrib search modules that might allow filtering to work.

Can I override the results page with a view and filter there?
With a View you will need to do the following: 


Make sure, you have also a second language and content translation is enabled (else you won't able to see the filter we need)
Add Filter Criteria: "Translation language"
And then check the following checkbox: "Interface text language selected for page"


Now this view is only going to list elements, which are available on the current language (and their translated version is going to be listed).