Cub 582, with 16hp Briggs & Stratton
 

When I go to connect the battery, as soon as I touch terminals, the tractor starts turning off, without the key being on. What would be the cause?

Confusing.... did you mean "turning over"? I assume this happens without proper safety switch protocols (driver in seat, pto off, brake depressed, etc...)

Unable to access manuals to see what you have besides knowing you have alternator from TractorData......so my GUESS is your starter solenoid is either stuck or hot (live). If hot, possibly a short in wiring or on key switch connections. Or maybe a bad key switch (internal) . You should be able to trace problem with meter without battery connected.

Plus, there are two different wiring for the 582, depending on serial number range.

:Huh:

Plus, my ol' lady hid my crystal balls so how 'bout some history on this thing!
Like was it working fine got for a long time and all of a sudden now you got this problem,
or did you replace some parts and now you got this problem,
or you picked up a non-running pos and this is what somebody passed on to you?
Until you provide details you can't expect any accurate help.