Hey Jake!
Victor from Shopify here :)
I do know of a few Dropshippers that have a direct integration with Shopify, though none of them are meeting all the criteria you have laid out, however you are not going to be limited to just 1 dropshipper for you store - you can use multiple dropshippers for different product ranges. Here's 1 for example, they have a decent number of Household products and a good Bulk selection as well: http://sunrisewholesalemerchandise.com/wholesale-products/Household+Items
Here's 1 for K cups: https://www.alibaba.com/trade/search?fsb=y&IndexArea=product_en&SearchText=keurig+k-cups+dropshipping
Finding one that can ship 2-3 days would be a challenge as most Dropshipping companies are overseas, but I'm sure if you find one within the U.S. this wouldn't be an issue. :)
Hope that helps!
Victor M