I used a stick blender for my first and only batch because I had one and I could. But people have been making soap for a long time without them. I've read recently that quick, consistent stirring is important. The same article or blog post or whatever said that you could use a stand mixer for small batches. I don't see why a hand mixer wouldn't work, too. But I think I would use one dedicated to soap or crafting at least, and not the kitchen mixer for making cakes. (My stick blender is cracked and on it's last leg. I need to replace it, anyway.)
Also, I used a large round, blat bottomed plastic bowl for my first and only batch of soap. It was a half-batch and the bowl worked for me. I didn't line it, just stuck it in the freezer for a few mintues before 'unmolding' onto a wooden cutting board. Then I cut the soap and put it back in my pantry, still on the cutting board, to cure. Insulating wasn't difficult. I just used a couple of bath towels. If you used a styrofoam cooler, you would need to line it so you wouldn't have to bust up the cooler to remove the soap.
But all of this is just my experience.
