I would try reinstalling the OS with the install disks, then from Finder, go into Applications > Utilities > Migration Assistant > and choose to restore from time machine backup. If the same thing happens, check check the minimum RAM requirements for your chosen OS, and also install SMC Fan Control. I've seen some strange things solved by SMC fan control--especially after an SSD upgrade.
One preliminary test is to plug the phone into a computer and see if it's recognized; if the phone is recognized, then it's most likely not totally gone. From what I've heard, this phone is an entry-level smartphone, meaning that they could care less about the durability of their parts. This means that a new screen should do the trick, providing previously mentioned test worked. I am not sure about the price of the parts, but for these niche smartphones, ebay or Amazon are usually the only places to purchase the parts. The other option is to take it into a repair shop and have them buy the parts and perform the service. Best of luck!
The standard ones should repopulate when the device is restored. Maybe try restoring again? Or, does your device have a file explorer? if not, you could try downloading one, (like ES file explorer) and look around to see if there is a media folder containing those audio clips. Worst case scenario, you can download a ringtone app from the Play Store and it will create a new ringtone/audio directory with some audio clips. Those new clips should then show up in native phone apps like "phone", "clock", etc.
If you do go ahead and attempt to clean the phone yourself, be sure to remove ALL of the metal shields on the logic board. Many people overlook these when cleaning up liquid damage. There are three or four of them I think. They are soldered to the board, but they can be pried up with a small flathead or tweezers, so long as extreme caution is taken to not damage components underneath the shield. And as previously posted, some companies offer iDevice data recoveries (Drivesavers comes to mind) where they actually extract the memory chip and access the raw data in another device. Fair warning, this can be quite costly. Best of luck!
There are quite a few ways to go about it. I have a PC, and Media Monkey is by far the best tool for the job. Just select your device from the list, then select which songs you want transferred, and simply drag them from Media Monkey over to the iTunes window. (I haven't used it in a while since I got Spotify so the directions may have changed, but it should pop up right away on Google if you search it). This is by far the easiest method I've used since it uses a graphical interface and organizes the media by song/artist/genre/etc. ,versus raw the raw filesystem, which can be a mess.
That does sound like a logic board problem. Unfortunately, Costs to fix such problems are usually close to--or more expensive than--replacing the phone. Have you considered selling the broken unit on ebay for parts and putting the profits towards a new/used phone? The market on ebay for broken devices is generally very attractive.