definetely a dab on the brakes after take off to stop them spinning and ensuing precession
This is something I recommend against. That spinning tire has increased it's diameter a little due to centrifugal forces, and is not being pressed into the rim by the aircraft weight. Too rapid application of the brakes can stop the rim, but the tire not so fast, and shearing off the valve stem is a possibility. Let them slow down on their own.
hypothetically certified with Tundra tyres the demonstrated crosswind limits
I don't know a lot about the thresholds of directional control certification. I've only done certification test flying on three taildraggers. The Beaver and C 185 were very forgiving, and met the requirement, so I was not asked to demonstrate more. I did not get into "challenging" territory with them. The Siai Marchetti was a different beast entirely, but was not being "certified" so I could pick my points. I did not pick crosswinds!
Are these differences less noticeable on a tricycle?
I would expect so, but I have never flown a tundra tire tricycle. There are much less common, as they are just not a great idea. If the ground is so rough/soft to require tundra tires, you're probably going to rip the nosewheel right off! It's also hard to find a nosewheel fork big enough to reach around.
The closest I have seen was this C 206 in Tuk, Alaska. Note the "mud" flaps on the main wheels to prevent gravel damage of the H stab.
