Should I replace my Macbook battery, charger, or both after 5+ years of use?
If so, what should I replace in order to increase my MacBook Pro’s battery life?

Emphasis mine
The battery.  Period.
I would replace the battery only if having 2-3 hours of usable charge is no longer convenient.  The question you have to ask yourself is whether or not the battery, in it's current state is still usable to you.  If it's not, change it.  If it is, try to get every charge cycle you can out of it (I'm a big proponent prolonging the life of products to minimize e-waste.)
You might find this post quite relevant:  Quality factors relevant for getting a replacement battery (MacBook Pro (13-inch, Mid 2012))

Battery Life

My battery over the years has underwent 2000+ cycles, and can only last at most 2-3 hrs from 100% to 0% currently

You are definitely several standard deviations from the norm on the right side bell curve that represents charge cycles and battery life.  Batteries generally have a useful life of 3 to 5 years. The &quot;1000 cycles&quot; is just a marketing estimate of how many types you can charge cycle the battery so you can estimate (budget) for battery replacement.  It's not a hard and fast rule (unfortunately, too many people believe it is).
MagSafe Adapter

My 60W MagSafe 1 charger (L-shape) has had some parts of the outer cable torn (half of it covered in electrical tape). The cable has become more and more yellow, and the plugs are slightly bent out of shape, although it still works.

If your charger still works and is safe you have no reason to change it.  Yellowing of the plastic has nothing to do with performance - it has to do with UV exposure to the plastic.  If you're still making the connection with the adapter, there's no reason to change.



NOTE: If it's brown, that means it's overheating and burning the plastic - destroy that charger immediately and get a new one.



I emphasize safe because the ground wire in the MagSafe cables is also EM shielding.  If this wire is broken you should replace the adapter.   Here you have a couple of options.

If replacing the adapter, I would get the 87W adapter.  It will give you more options if you choose to add another vintage Mac to your stable.  You can always use a &quot;larger&quot; charger than what your device is rated for; not smaller.  And with the MagSafe 2 Converter, you can use that MagSafe 1 with newer devices!  See Apple's Support Document Find the Right Power Adapter for Your Mac for full details.

If you have the technical skills, you can &quot;break&quot; open the MagSafe adapter and simply solder on a new cable.  Yes, you have to break it open because it's glued together (stupid design), but once open, you can solder on a new cable and (what I did) duct tape the adapter back together. I have one still working to this day.  iFixit.com has excellent guides on how to do this.