Diablo – the “Item Budget” for Blacksmith created gear

You can't

There is no search feature built in, however your items are sorted by item type first, and then by required level (highest level on top)

So to find an item, simply go to the item category (Bracers, Crossbows, Shields, etc), and then the approx level of the item you're looking for.

My observations in the Auction House do indeed show there is server-side caching going on with requests. My completely uneducated guesstimate is it will refresh your cache approximately every 60 seconds. However, there is a way around that:

Let's say you're looking for a cheap item with a buyout of less than 5000. Do your first search on 5000. Then re-search with 5001. Then 5002, and so on. The reason why I say this works is researching the same amounts over and over again always return the same response through the first 60 seconds or so. If I change my max buyout amount, I WILL find items listed that I didn't see before that match the previous criteria (i.e. change my max buyout from 5000 to 5001, and an item with a max buyout of 1499 will show up.)

Hopefully this helps.

EDIT: I realized on re-reading I need to clarify something. What caches with the AH is the specific items listed, not the time left in auction or the ability to buy them. When you refresh the identical query over and over again, if the item listed sold it will change to "Sold". The time left in the auction will appear to be the same. However, the query will not search for new items that meet your criteria, which is what the question is referring to.