I wonder how I can view how many downvotes I received on my questions across the SE network.
I'm not really sure what you want to do with that information. Personally, I'm more interested in how many people I helped, so I'd rather look at the number of upvotes and accept votes. Fortunately, SQL supports pivoting, so we can have a look at everything at once. The number you're looking for is 2328 (well, it was last Sunday morning; the downvotes on this question (not mine) haven't been stored in SEDE yet).