Can I jailbreak my way out of a passcode request? My father owns an iPad 1. He forgot his numerical access code and tried enough times that the device now reads "Connect with iTunes."

Apple assumes you sync your device frequently with a computer  but the only time I recall him doing that is when he upgraded to iOS 5. iTunes on that computer, however, says "no can do, put in the code first."

Apparently the only way out is to boot in recovery mode and flash the device clean, which means losing permanently access to all apps ever, backwards and forwards in time. (We have been unable to install any new app at all for a year or so.) This means being stuck with a version of Safari that, as you well know, runs out of memory and crashes and burns on most websites.

Hooray.



Can I jailbreak the device now? Would that help me get out of this situation?
No.
You will need to restore using iTunes. The iPad will completely be factory reset (meaning like it's fresh out of the box). Your personal data on the device will be lost but you can redownload all apps for free using the same Apple ID you bought them with. 

However you might run into some issues since the iPad 1 only runs iOS 5 and most apps require iOS 6 nowadays. Some might still be downloadable since Apple added a feature to support old iOS versions parallel (see http://9to5mac.com/2013/09/17/ios-app-store-now-allows-legacy-app-downloads-to-support-older-hardware-and-software/). However this will depend on the App Developer, so you might run into trouble getting some (probably a lot) of Apps back if they are not backed up to your computer.

Safari isn't an App that can be updated by itself, it comes with iOS, and the version depends on the iOS version.