Very nice find. You need to show us a photo of the drive side or at least tell us what rear derailleur you are using to be able to determine how large a rear cog you can use. It looks like that the bike has a short cage "racing" derailleur and that you'll need a different one to get hill climbing gearing to work.

The size of rear cog has no effect on the susceptibility of the rear axle to breakage. That is more a function of dropout alignment, rider weight and how many potholes/curbs you run over. If I were you I'd stick with a freewheel setup since it's the cheapest way to go and should work fine if you don't spread the dropouts. If there's a bunch of extra space between the stay and your current smallest cog you might be able to fit a 7-speed in there...that's what I did on my Miyata. Of course, if you have index shifting you won't be able to do that and will need to stay at 6-speed.