Your most recent choices will always be remembered for next time. To save multiple searches, you have a couple options.
As long as you stay on the same computer and don't clear your browser files, you can pull up your recent searches using the Your Searches button at the top-right of SearchTempest. That button also allows you to set favorite searches, which will be available even if you haven't looked at them in a while.
To make sure a search is kept forever, just bookmark the results pages in your browser. When you open the bookmark, the latest results for that search will be shown.
If you want to save a particular ad you've clicked on, make sure you get the link for the ad itself, not the link to the whole SearchTempest results page. The easiest way to do that is to middle-click (Cmd+Click on Macs) on the ad link to open it in a new browser tab.