For example I would like to search the web as if it was 2010 and only results from 2010.
They are stored on the archive.org, so I think something like this should be possible.
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Probably not the way you're thinking, but you can search for results and limit by last update date.
After performing your search, open up the "Search Tools" menu under the search field.
Open the "Any time" menu and enter a "Custom range..." (I'd expect 1/1/2010 and 12/31/2010.)
That'll give you results from the Google index. Once you've found a page you want to look at, you'll need to open it in the Wayback Machine.
and then choose a date in 2010.
You can limit the Wayback Machine to only return results from a particular year (or date) be including that in the wildcard:
Note that Google might be wrong about the published date, and something that may have existed in 2010 but doesn't exist now may not be in Google search index at all. Better would be if the Web Archive included a robust search engine for the Wayback Machine data, but they don't. (It's probably resource-intensive and cost-prohibitive.)
Web archaeology is hard.