Mark Smith, a former railway worker, runs a guide to buying tickets and planning train journeys across Europe and the rest of the world.
HN commenters recommend it as the place to check for detailed train travel information and how to buy tickets between almost any two cities.
Eurostar runs direct trains from London to Rotterdam & Amsterdam, 3.5 and 4 hours respectively.
You can't really compare to a similar route in the US because there are hardly any and hardly anybody uses them. The train routes in EU are busy and frequently full, just like planes. Obviously booking ahead is cheaper, how else could you do it?
Top tip, plan a head and read up a little bit on seat61:
https://www.seat61.com/trains-and-routes/london-to-amsterdam-by-eurostar.htm
Top tip 2:
Overnight trains is one of the best ways to see Europe as a backpacker on the cheap. The routes are often scenic and the cost is frequently cheaper than anything else since you don't have to pay for accommodations.
Buses are a thing too. And they are not greyhound bad, but definitely not as good as trains.
I was speaking more generally, sorry; there is currently no daily Paris-Berlin night train (and I guess the weekly Russian train is suspended). The most reasonable connection seems to be: Paris 19.04 (ICE 9557) 22.17 Mannheim 23.47 (Nightjet 470) 07.38 Berlin, with some time to drink a nightcap in Mannheim, but I can see how that won't work for everyone. You can also leave Paris at 17.55 and change more comfortably in Karlsruhe with maybe a late dinner. In the reverse direction you'll have an unpleasant change in the early morning though.
In daytime, there are a few options with only one change, but you'll sit in a train for 8 hours or so, which may not be your thing.
In general, https://seat61.com/ has good tips.
That'll happen if you run off the end of the connections that CD can book for you. For example I recently tried to book one all the way down to Trieste and it could show the trains but couldn't book them for me, you need to break it down into parts and book with relevant operators, in my case CD for the first part and Trenitalia for the last, skipping OeBB in the middle. Just beware when you do this that different operators will show different trains and connections so you may need to manually select the trains if (for example) CD and Trenitalia disagree over the best links.
Seat 61 has an excellent guide on dealing with this, https://www.seat61.com/european-train-tickets-online.htm.
Will check it out. Until last year, you could use a neighbor country to buy train tickets to and from Germany. Since you did not have to start using the ticket in the first station but can basically board the train wherever you feel on the booked trip, you were able to buy all bullet train tickets for about 20 Euros (e.g. Berlin>Hannover, Leipzig>Cologne etc.). Best thing: Not bound to a specific train on that day and they were reimbursable until 24h before your trip started. Good times. But all good times come to an end :-(
Just for a comparison: These 20 Euro rides would set you back 150 Euros otherwise.
And for train ride information, there is always: https://www.seat61.com/
Amazing writing, I couldn't put it down. Don't miss the Amtrak Tips page linked at the bottom.
For anyone fascinated by rail travel, I can recommend the train from Bulawayo to Victoria Falls (Zimbabwe). It's definitely like he writes, hard mode, but it's only for one day or night, and when I was there first class tickets cost $7 US.
Also worthwhile: South Korea (trains galore, hard mode only due to the writing system). Sri Lanka's Hill Country (the train goes about 10 kph and you will wish it would slow down--the views!). Oslo-Stockholm (easy but expensive mode).
And finally, http://www.seat61.com - OK, it's got ads now, just ignore them, this is a gem of Web 1.0.
The emissions of trains are always many times less.
For a quantitative comparison:
https://www.seat61.com/CO2flights.htm
For example:
London to Nice
Plane: 4 hours, 250 Kg CO2
Train: 8 hours by Eurostar+TGV, 36 Kg CO2
85% less
Traveling short to medium distances by plane is simply indefensible if you care about climate change. For an average European, it can easily make up one third of their total yearly carbon emissions, more than 1500kg, or even more.
(BTW this site, seat61.com, is a great and very useful web site, if one likes train travel or wants to travel by train in Europe, one should seriously have a look!)