Now that the newly posted question How to find questions with very negative votes? has been marked as a duplicate of this question. I would like to post my updated answer for this question specifically addressing the following two questions asked by OP:
OP's question in the title says:
Is it possible to search for down voted questions?
So, as per the question OP wants to find the down voted questions and the answer for that is to search it by entering either of the following two search options in the search box:
The above search query will only display all the questions with negative votes. It's very important to include the is:question type option here otherwise just searching with "score:..-1" (as given in the OP's own accepted answer) would result both i.e. questions and answers with negative votes.
Here is a screenshot of result shown using just score:..-1 (top section of the 1st page of the total 9,920 results shown)
If you notice, the last two posts have "answered" in the status posted by user's BalusC and CanSpice respectively. If you click on them, you will see that it's the answer that got the negative vote here whereas the question has only up-votes (and no down-votes).
Here is the screenshot with a search query using score:..-1 is:question option: (notice 7,375 results as oppose to above 9,920 results)
The above search results only displays all the questions with negative votes and have a question status of "asked". So, it is very important to include the is:question type option here.
Now, lets move to the next part of the question. In body of the question OP says: (emphasis mine)
I want to search for new downvoted questions to improve or flag them.
For this, all you have to do is click on the "newest" search tab and search result will be sorted accordingly.