Q: Latest update 7.1.2 stuck with less than one minute remaining

There are no absolutes guarantees, but you should not loose your apps in this situation. I have recommended the reboot to more than a few users when the update stalled like this, and it worked the overwhelming majority of the time with no data or app loss whatsoever. But like I said, there are no guarantees. If you sync with iTunes, the apps should still be in your iTunes library on your computer.

Besides, you are between a rock and hard place anyway, as the saying goes. Unless you do something to get the iPad working again, you may be able to keep all of you apps -  but you will have a device that doesn't work anyway.  So what good is having all of your apps if the iPad doesn't work?

Bottom line, I think you will be fine. You are basically force restarting the device when you reboot.

iOS: Apple logo with progress bar after updating or restoring from backup

http://support.apple.com/kb/TS3681

If you can't download apps, it's probably because you don't have sufficient storage space. You need to increase your Available space.

iPhone, iPad, and iPod: Understanding capacity

http://support.apple.com/kb/ht1867

How much space is used by your Other? You may be able to reduce.

How Do I Get Rid Of The “Other” Data Stored On My iPad Or iPhone?

http://tinyurl.com/85w6xwn

How to Remove “Other” Data from iPhone, iPad and iPod Touch

http://www.igeeksblog.com/how-to-remove-other-data-from-iphone/

With an iOS device, the “Other” space in iTunes is used to store things like documents, settings, caches, and a few other important items. If you sync lots of documents to apps like GoodReader, DropCopy, or anything else that reads external files, your storage use can skyrocket. With iOS 5/6/7, you can see exactly which applications are taking up the most space. Just head to Settings > General > Usage, and tap the button labeled Show All Apps. The storage section will show you the app and how much storage space it is taking up. Tap on the app name to get a description of the additional storage space being used by the app’s documents and data. You can remove the storage-hogging application and all of its data directly from this screen, or manually remove the data by opening the app. Some applications, especially those designed by Apple, will allow you to remove stored data by swiping from left to right on the item to reveal a Delete button.

What is “Other” and What Can I Do About It?

https://discussions.apple.com/docs/DOC-5142

iPhone or iPad Ran Out of Storage Space? Here’s How to Make Space Available Quickly

http://osxdaily.com/2012/06/02/iphone-ipad-ran-out-of-available-storage-space-ho w-to-fix-quick/

6 Tips to Free Up Tons of Storage Space on iPad, iPhone, and iPod Touch

http://osxdaily.com/2012/04/24/6-tips-free-up-storage-space-ipad-iphone-ipod-tou ch/

Also,

How to Clear Message/iMessage Cache on iPhone & iPad And Reclaim Lots of Free Space

http://www.igeeksblog.com/how-to-clear-message-imessage-cache-on-iphone-ipad/

What is Stored in iCloud and What is Not

https://sites.google.com/site/appleclubfhs/support/advice-and-articles/what-is-s tored-in-icloud

 Cheers, Tom