2002 ZL 600 EFI ESR Starter

My son bought the above sled. The dealer said the battery was shot,
so I had him try using a battery pack to test the electric starter.
It seemed pretty anemic and wouldn't spin the engine over at all.
It appeared when the starter drive gear engaged that the gear
would slip on the flex shaft.
I thought I read something on the Net that part of this system
had an upgrade. Can anyone tell me about this, or what the problem
may be?
I pulled the battery to check it out, and the case is cracked in two
corners, and it has no fluid in it. Shot for sure. Therefore, have
not played with the electric starter anymore, until he gets a new battery.

Also, it started at the dealer normal, on the third pull. Once we got
it home, and the half dozen times it's been started since, it starts
on the third pull or so when cold, but runs at an extremely low idle.
Sometimes it dies after 20 seconds or so, then on restart, idles normal.
Not much snow yet to test, but seems to run pretty good after.
Not sure if it's only firing on once cylinder when it's idling extremely slow.
Will try new plugs, but thought I'd ask it there's other possible causes?

Also, I see (when I had the battery out) that the starter solenoid
is mounted below the battery box, and has some corrosion on it from
battery acid. I'd like to get it out and clean it and the cables up, but
see no good way to get at it, and the battery box seems to be
attached at many points. What's the best way to tackle it to remove the
solenoid?
Many thanks for your help.

2002 ZL 600 EFI ESR Starter