How do I ensure that the queries I type in the search window are remembered so I don't have to re-type them?

Answer 1

What you're talking about is "Auto Complete"
To access this, go to: Internet options>Content>'Autocomplete settings' & select the stuff you'd like 'it' to remember!

Answer 2

Just don't delete your history. You can set the number of days for history to remember by going to:
Tools
Internet Options
Next to the delete button, is a button for "settings"
click that to set your days to save