How do I prevent the Browser and Dolphin HD from deleting old history entries?

Dolphin Settings

Well, for starters, I'm looking at my own Dolphin Browser Settings page right now and under Data settings, you have the option to manually clear certain data types like history and cookies and also have the option to manually back the data up to your local or external (SD) storage disk.

Data Backup

In addition, under Sync settings, you can actually backup an unlimited amount of your history, bookmarks, cookies, etc. to your Dolphin Connect Account (which you can connect to Google or other social services). So that may be an option you may want to look into, is data backup so you'll actually have access to unlimited amount of the data you use from any device you use Dolphin on, despite Dolphin's local limit.

Drawing The Line

As far as I'm concerned (and someone, correct me if I'm wrong), at this stage I don't know that there is an option for you. The new Dolphin Browser doesn't have any limit so I'm not sure how much assistance any one of us will be knowing that Dolphin HD is an outdated version of Dolphin which is no longer supported.

The new Dolphin Browser can be found here.

Good luck and let me know if I can help any further! I'll continue looking in the meantime, but it is doubtful that I'd find much.