The second bit may be it. I usually let the nose come down on its own, and be sure when you put your feet on the brakes the pressure is even. All the Cessnas I've flown besides the 210 had hyper sensitive brakes.
Tordon, Tupac I think it's good airmanship to 'land' the nose wheel, but that's just my experience, but I witnessed many pilots that don't. I say that because I flew with Metro guys that do a fantastic main wheel landing then just drop the nose hard on the ground then jump on the beta ... you feel all the weight transfered on the nose wheel ... and man
that can't be good for that nose strut.
I usually touch main, hit the beta, land the nose wheel and maintain as much back pressure as possible till the elevator effectiveness dies around 50 kts. You really don't feel the nose sink like it's trying to go thru the ground. But that's me in this plane, and also the 402s and other bug smashers. Maybe jets you don't care, can't afford to do it.
As far as crosswind, we have no Ops Specs limits as far as Xwind, we do have 10 kts tailwind. And thank God, those Idaho airports get hammered. Xwind landing on a dry runway is not too hard because we have a very responsive differential power/beta to keep the plane in check. When the runway is iced up it's even easier, you can land with the crap as long as your 'momentum' is lined up with the runway, then you manage with differential beta.