Lotsa questions. Let's see if I can take them one at a time.
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Do I cut the soil pipe off at the concrete slab surface leaving no pipe extending above the slab?
Yes, Unless you're going to tile Then cut level with the tile floor.
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Should I install the closet flange directly to the concrete slab and if so, can I use something like a tapcon screw to secure the flange to the slab?
Although you didn't say I'm assuming the pipes are PVC and not cast iron. If you're going to leave the floor as it is you will prime and glue a PVC closet flange directly to the floor. The glue joint shoulod be enough to secure the bole but if you're concerned then you can use TapCons to secure it more.
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If I install the toilet to the closet flange attached to the concrete slab do I need to put something like thinset under the closet flange and toilet to act as a barrier between the toilet and concrete?
No!
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What type of adhesive do I use to adhere the closet flange to the soil pipe?
You will use ABS or PVC Cement and Primer depending on the type of pipe.
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When/how would I use plastic plumber wedges?
Plastic wedges, called shims, are used if the floor's uneven. They are used to keep the bowl from rocking.
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If you can help me, please provide a step-by-step process so that I can get this toilet secured to the floor.
Setting a toilet is a simple job. Put the closet bolts in the flange slots. Now place the wax seal on the bottom of the stool and set the stool over the flange so the bolts come up through the holes in the base . Give the stool a twist to seat the seal. Now is the time,if you have china caps, to put the plastic locking wafer on the bolt first, Making sure the beveled side is down, now the washer and nut. DO NOT OVER TIGHTEN as you can break the china. If you tighten it so that you can't twist it it's tight enough. Tie in the water and test for leaks. Final step is to take a small tube of DAP tub and tile caulk and grout around the base. Clean the excess off with a damp cloth. Congratulations, you have just set your commode. Hope this helps and thank you for rating my reply. Tom