I don't see any major correlation to fixed gear in this situation. It sounds like a crap tube or at least a tube that could have used the valve core tightened. Get yourself a valve core tool like the Park Tool VC-1 and start tightening down your valve stems. Also maybe get a different pump that has a better head that won't loosen valve stems because that could also be the culprit. You can now get Gatorskin Tubulars so you have flat protection and a tire that won't come off and with proper inflation you are less likely to encounter flats.

As far as dropping a chain, Aaron Gwin won at the UCI World Cup DH finals chainless in 2015. He broke his chain out of the gate and still beat everyone else. I am not saying we are all Aaron Gwin or could reach his level but if he can do it on a world cup downhill course I think we could do semi ok on a smoother road.

Also as TimH said maybe a rear brake is in order as well. If weigh is an issue eeBrakes from Cane Creek with Elite Link housing and say SRAM S900 carbon brake levers (assuming a drop or bullhorn style bar) and if cost is an issue I bet your local co-op will have a rear brake laying about or you can get some cheapie Tektro or even Shimano Claris and if you spend a little money on high end brake pads you won't be so bad off and you could take a shifter and gut it or buy some cheap Tektro levers. If it is aesthetics so many to choose from but Paul Component Engineering usually dose some of their stuff anodized in different colors like the Racer brakes, Ultegra or Dura Ace always have looked nice and with those TRP RRL levers or even the SRAM levers are nice looking or for that vintage look Shimano still makes something decent. Heck you can even get hydraulic rim brakes from Magura granted they are for TT basebars but I bet with some hoses and fittings you could make them work with say a TRP Hylex lever since both use mineral oil. (but don't quote me on that)