I understand your concerns - I thought the same when I got the solder pot I have. It looked tippy, because it was kind of tall for it's breadth. However, the base had enough ballast to make tipping a relatively low risk. I placed it on a piece of ceramic tile, thinking at first the based would get hot enough to overheat the benchtop. It doesn't, but I later decided the tile was good insurance in case of splashes or a tip-over. I suppose one could somehow attach the thing to a broader base and make that practically impossible, but in my experience, that didn't seem necessary.
I do like that the Hexacon style has a small, narrow reservoir, as it seems less likely that things might get dropped in it. I don't think you'll be disappointed - it does get the job done, and I've had no problems using mine. Also, it doesn't take a lot of solder to tin wires, so there doesn't need to be much in the pot.
As to what kind of solder, I bought typical bar solder (see link), most likely 63/37 Sn/Pb. I bought a half-dozen 1 lb bars, which should last me about 1,250 years. As I recall, I cut a few chunks off the end of one of them, put that in the pot, and have never even had to add more. I haven't tried any lead free solders, much less anything for the pot (especially since it's still got a plug of solder in it, and I have another 1,248 years worth somewhere in the shop) so I can't offer any useful advice there.
http://gokimco.com/kester-04-6337-0050-ultrapure-bar-solder-sn63-pb37-price-per-lb-sold-by-bar.html?___store=default&gclid=CIa6y4fpxroCFQVBQgodPQQAeQ
