Q: Random Rests - 1st Generation iPod (2001-mechanical wheel) Classic

Hard to believe the battery still holds a charge, but the 1st and 2nd gen iPods DO have physically HUGE batteries compared to later models, and they are lithium polymer (versus lithium ion), so maybe those factors makes a difference.  Plus your "daily use," which is better for a battery, compared to sitting unused for long periods.

Have you (ever) done a Restore on the iPod using iTunes?  In this case, I actually HOPE your profile info is up-to-date and you use an old PowerBook with Leopard, with a REAL FireWire port.  This erases the iPod, reinstalls its old but latest software (which should be version 1.5), and resets it to default settings.  If your version is iTunes 10.6.3 (the last version for PowerPC Macs running Mac OS X 10.5.8), you should be able to connect to Apple's servers and download the iPod's software package (if not previously downloaded).  That older version of iTunes is less likely to cause problems doing a Restore on your iPod, although it should still work with the latest version of iTunes.  The Restore button is on the iPod's Summary settings screen in iTunes.

If you've never done a Restore during 15 years of ownership (which would also be amazing), it gives the iPod a fresh start, at least for its onboard software.  If the problem recurs after doing a Restore, the cause is more likely to be a hardware problem.

Does this "random reset" problem ever occur when you play the iPod while it is connected to the FireWire power adapter?  If not, the cause could be related to its old battery.  It may still hold a charge, but it may be worn out and getting weaker, and not supplying enough ongoing power, which causes the random resets.

I just recently took the iPod into the apple store... otherwise I'm at the mercy to apples "genius bar" for a replacement battery.

The Apple Store will probably not be able to help.  Your iPod is about as "vintage and obsolete" (Apple terminology) as an iPod can get...    Unless you happen to speak to a Genius with personal knowledge about really old iPods, but that Apple technician may not be allowed to work on it (by policy).

Vintage and obsolete products - Apple Support

I do have a 1st gen iPod.  Until recently, it had a non-functional scroll wheel (so it was sitting unused), but I fixed it by "swapping parts."  I also replaced the battery and substituted a 20GB hard drive (that came from 4th gen iPod).

I don't think listening to music while charging simultaneously is possible on this iPod.

It works on mine.  Since you have never done a Restore on your iPod, you may have an older version of the iPod's onboard software.  The last version is 1.5.  Unlike recent iPods, where the onboard software does not change very much for a given iPod model (just bug fixes), the early iPods had significant software updates.  For example, I don't think the AAC audio format even existed when the first iPod was released, but it plays AAC just fine (including the "protected" variety).  So that may be another benefit of doing a Restore.

Does your PowerBook run Leopard (10.5.8)?  If so, the latest version of iTunes is 10.6.3.  Another user posted that he could not download iPod software packages from Apple's servers when trying to do a Restore using an even older version of iTunes.  When he updated iTunes to 10.6.3, it worked.  I have a Power Mac G5 running Leopard with iTunes 10.6.3, and it can connected to Apple's servers to download iPod software packages (it did a few weeks ago).

If you need to replace the battery at some point, this iPod is trickier than others to open, without causing unintended damage.  But once opened, it is easier, because the components are relatively "huge."  The major components connect with actual connectors (not solder).  There are no ultra-thin and delicate ribbon cables and connectors.  You can post back if you want some tips.  This web site also has illustrated repair guides.

https://www.ifixit.com/Device/iPod_1st_Generation

Since your battery shows odd behavior, that is a likely reason for the iPod resetting unexpectedly; it has lasted well beyond expectations.  Another possibility is the hard drive becoming unreliable.  And it could be data corruption, especially since you have never done a Restore before.  If it's only data corruption (no hardware problems), the Restore will resolve it (by erasing the hard drive).

The battery part is getting harder to find.  Here's one on Amazon, that's described as a higher capacity battery.

http://www.amazon.com/2200mAh-Battery-iPOD-Generation-Extended/dp/B00G5V0XYQ

And it's also listed on Apple-centric vendor OWC's web site.  The OWC price is higher, but it says you get a pair of the plastic (nylon) pry tools that you need to open the iPod without causing ugly damage.  Plus they kindly produced a HOW TO video (a good one), which is linked to the product's description

http://eshop.macsales.com/item/NewerTech/BIPOD12HC/

If you order from OWC, you may also want to add this special tool, which I've found to be generally useful when working with iPods.  It is made of metal, and is somewhat sharp (so be careful); it can scratch the iPod's surface, but it may be helpful in getting the seam initially apart, so you can insert the "non-scratching" plastic tool into the seam (to work around the iPod as shown in the video).

http://eshop.macsales.com/item/NewerTech/TOOLISESAMO/

The repair guide shown here is also clear and well-illustrated

https://www.ifixit.com/Guide/iPod+1st+Generation+Battery+Replacement/365

Here are my additional tips.  On the top edge, which is the edge with the FireWire port and headphones jack, it is mostly plastic.  Do not pry apart the case along that edge.  It's tempting, because that edge is the easiest place to initially insert the plastic pry tool.  However, doing so can cause the plastic surface along that edge to crack.  It does not crack and fall apart, but it looks bad.  Open the seam along the long edge and go around the bottom edge of the iPod, as shown in the video and repair guide.

Gently separate the battery.  That "iSesamo" tool may be useful here.  Those adhesive rubber strips can be REALLY sticky after all these years.  You don't want to be too rough with the hard drive, which is what the battery is stuck to.  After disconnecting the battery, make sure the hard drive is still firmly connected to its ribbon cable. 

After installing the new battery, charge it up and use the iPod without closing the case completely.  After confirming everything is working fine, close the case completely.  That's about it.  Not very difficult (after getting the case opened).