You'll need maybe at least a 7-8 gallon mash tun capacity for high gravity 5 gallon batches, I think. Possibly more if you prefer thinner mashes or depending on exactly how high a gravity you are talking about.
IME, it's worth spending some amount on a decent false bottom and valves whichever way you go. That can easily add up to as much as the vessel itself. So make sure you are comparing total costs for both. I don't think it's worth trying to hack together the various braid or manifold type systems, as stainless steel false bottoms are relatively cheap and easy to use and to clean. Similarly for adding ball-valves - it's probably cheaper in the long run to buy a stainless kit from someone like brewhardware than to spend significant amounts of time hunting for the right parts to jerry-rig something together at the hardware store.
I used a 10 gallon round cooler for several years, but recently the wall collapsed where the outlet goes through, and started leaking. I've shifted over to my old brew kettle, but I'd already set up a RIMS system, so I'm not so concerned about insulation. But I think it wouldn't be too hard to insulate a stainless steel mash tun relatively well. If you can see yourself going to a RIMS or HERMS type system in the not too distant future (e.g. to play with step mashes, mash outs or control mash temperature more tightly), then insulation of the mash tun could be done on a more temporary basis, and going stainless now would make more sense than it might do otherwise.