$6k is reasonable for a 2011 with a good-not-great battery and that odometer reading, especially if it's cosmetically nice. $5k might be closer if the car has dings and upholstery damage. This is all subject to things like EV popularity and availability in your area.
Welcome. Yes, there is a starting type 12 volt battery in the Leaf, along with most other EVs. If there is an outlet nearby, your best option is to connect a 12 volt external battery maintainer, leaving the battery installed in the car and connected.
People have installed lithium iron (LiFePO4) batteries, with good results, in warmer climates. Most replacements, though, if they aren't OEM type flooded cell batteries, are AGM type, which work better in the Leaf than the OEM.
This sounds like one or more cells in the battery are going South. You need to use LeafSpy Pro to get a depiction of the battery cells and the voltage variations between them. Hawaii is kind of deceptive, in that while you don't usually get extreme heat, the generally warm/very warm nights prevent t...
What's happening is that Nissan, claiming to be short on 40kwh packs, is offering to buy the cars back. There is negotiation involved, so if they offer that, don't accept their first $ offer as their final one.