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Best way to tell is to jack both wheels off the ground & spin one by hand. If the opposite tire spins in the same direction you have a limited slip differential, if it spins in the opposite direction it's a standard or "open" differential. By the way, they're commonly refered to as *open* differentials because the differential case is "open" like what you were expecting to find. Now maybe someone else will jump in and correct me *again* <grin> but I think you may have out-thunk yourself by ordering the mini-spool first because it's always been *my* understanding that they pretty much only fit *open* differentials. You don't have an open diff, now maybe you can pick one up at a local boneyard for a few bux (I've done THAT before and it cost me $40). At this point you're probably best off either ordering yourself a regular spool or picking up a junkyard 3rd member w/an open diff. No matter how you cut it I'm pretty sure that what you have won't work with what you have.
No problem. All a spool is is someting that eliminates the differential all-together. You press the axle-shafts directly into the spool and it mechanically locks everythign together as if there was just a single axle shaft running from one tire to the other. This is kinda rough on roads because whenever you make a turn the inside tire covers less distance than the outside tire (a differential is what takes care of this in street vehicles & that's WHY it's called a differential because it it compensates for the *differential* in tire speeds between the inside & outside tire when you turn. Waay back in the day it was called a "compensating gear" because that's what it did, it *compensated* for the speed differential) A spool does NOT differentiate tire speeds so you'll always scrub a tire on hard surfaces. For off-road only trucks this is fine because the soft dirt or sand will allow one tire to slip and that will take care of the problem. A mini-spool does basically the same thing but it's smaller and can be installed into an open carrier differential saving time, hassles and $$$. A true spool will have the ring-gear bolted to it and the axle shafts pressed into either end. Spools are pretty strong, mini-spools work well in some applications but not all since they aren't as strong and use the factory open differential carrier, a full spool is usually much stronger but it requires more work to install.
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