Windows – Clear Windows “Run” box cache

I'm not sure how much relevance this has to your question, but to answer your 2nd part, yes, hard-drives have their own internal hardware cache - usually 2, 8, 16, 32 or recently 64 megabytes.

Aside from that, Windows has a feature called Disk Write Caching which may help you. It can be enabled/disabled on a per-drive basis doing the following (directly copied from here):

1. Right-click My Computer, and then click Properties.
2. Click the Hardware tab.
3. Click Device Manager.
4. Click the plus sign (+) next to the Disk Drives branch to expand it.
5. Right-click the drive on which you want to enable or disable disk write caching, and then click Properties.
6. Click the Disk Properties tab.
7. Click to select or clear the Write Cache Enabled check box as appropriate.
8. Click OK.

Following are the steps to clear history of the Run box:

1. Go to the Start screen, and type regedit and open click on regedit to open the Registry Editor
2. Navigate to the key: HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Explorer\RunMRU
3. On the right of the screen, there are keys that contain the recently entered commands on the Run Box. Delete the ones that you want to delete
4. Log off and Log back on.

That should delete history from the Run menu.

The UI Way to delete History

1. Type Clear History in the Search box. Then click on Settings on the Charms menu
2. Finally select Delete History to clear the history

However, that method deletes more than just the search history of Run Box. It also deletes history from other applications like File Explorer.