Most people use either old or fairly cheap (<€500) city bikes. We only lock the rear wheel; if you want to you can pick up the bike and walk off with it.
Some people do have nicer bikes, and they chain the bikes to something solid. I did this for 6 months when my city bike was new.
Sure, but how many will fit on one train/taxi? Here in Vancouver we have two bike-slots per bus and ~4 per train (which is actually just general accessibility-device space, for e.g. powered-wheelchairs, and is almost always taken already.)
Wikipedia says a suburban train takes 46 bikes. I should check, as the press release from the company doesn't give a figure.
I don't commute by train, so I don't know if that's still not enough. Many people do leave a second bicycle at the station.
https://en.wikipedia.org/wiki/Cycling_in_Copenhagen
https://www.dsb.dk/om-dsb/presse/nyheder/mere-plads-til-cykl...
I use mine for every trip longer than a 5 minute walk and less than a 45 mins bike ride. That encompasses the whole of Amsterdam and its outlying suburbs.
In bad weather, people have suitable rain gear and accept getting a bit wet. Or cycle with umbrellas. As the Dutch say, "there's no such thing as bad weather, just bad clothing".
But yes, the cycle paths are ploughed (often before the roadway). Nevertheless, bus and train use increases.
For cycling a long time in the rain, some people use overtrousers or other waterproof clothing.
(Also, the suburban trains and taxis carry bikes for free.)