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Absolutely. The stiffness of the bulk material won't be noticeably different between alloys, but the stiffness of the assembly will be - sometimes quite so - when tubing shape and size varies. And a stronger and tougher material allows bigger and therefore stiffer tubes with thin walls, this being a decent weight too.
Comments about thicker walls at joints for ease of welding in mass production environments aside, of course...
My project: I'm having a local builder create a replacement for my cross check, with some variations to geo and a few personalised touches. He's not got an online presence so I'll not throw names around here for the moment.
No it's a non issue (unless baths and electric current are involved) as far as bikes are concerned. SO long as appropriate welding/brazing rods are used there's no issues with mixing stainless and non-stainless.
Cool what's the build? Far and away the biggest component in determining the stiffness of a tube is the size and shape, I'd be very surprised if you could tell the difference between any OS tube sets of equal diameter. It's a while since I've looked through the Reynolds catalogue but if I remember correctly 853 only goes up to a 35mm DT so you may want to look at Columbus offerings. There's other reasons to choose a tube though and 853 is an absolute world away from anything Columbus make in terms of toughness. It may or may not make a difference on the bike but it's a lot more robust.