Advise On First 2+ Week Europe Trip

Six cities in 18 days is about 3 days per town. It's feasible, but if you are already concerned about being stretched too thin, then consider eliminating one or two cities.

Also, flying city-to-city may not be the time saver you imagine one you factor in security lines, flight delays, and transit to and from the airport. For example, you can get from London to Paris far quicker by rail than by plane. Depends on how you sequence the towns you visit.

Save Rome for a full trip to Italy.
Save Munich for a full trip to Central Europe.

Much more enjoyment, and a lot less stress here ...
Arrive London (4N)
Eurostar to Paris (5N)
Thalys to Amsterdam (4N)
Fly to Barcelona (5N)
Depart Barcelona

I concur with David that you should limit yourself to 4 cities
You need to decide which 4 are most interesting to you and then design the most efficient route for the ones you decide on.
And sometimes flying is not the fastest way

"Six cities in 18 days is about 3 days per town." Unfortunately, these are all nearly full-day travel days, so it're more like 2 days per town.

*In theory it's more like half a day to get to the airport with time to spare, wait about, fly, and then get to the city center. In practice with 4 friends traveling together it is a lot more. Just don't expect to see much on those travel days, except for a nice meal in the new city perhaps.

I agree you should limit the number of cities, but I would say do London, Amsterdam, Paris and Munich. Munich has a great Christmas Market.

Of course it is do-able, and if you and your friends want to see those cities then why not.
But I do agree with the others that it would be pretty hectic with all those cities and you may get back home and think what did I do there? One thing we learnt in Europe is that you need to take time out to enjoy yourself and immerse yourself in your surroundings as it is a beautiful destination...
When travelling Europe back in 2013 we stayed in all the major cities for four to six nights as there is a lot more to do, and then in smaller towns or villages for two to four nights.
Even if you took one city out of your list I think that would make a huge difference to the enjoyment of your holiday... But in the end, its up to your and your friends...

Theme emerging here, but especially for your northern three, be brave and use the train. It's faster, and far more enjoyable. Feels more adventurous too. Train from London to Paris and from Paris to Amsterdam is quick and high quality.

The southern places, sure you're looking at flying.

I agree with those suggesting maybe taking out a city, but in the end only you can make that choice. I get the temptation as it's not something that easy to repeat.

Since you are visiting in the Winter, days will be short and it will be cold. Only good news is there will be fewer tourists.

I wold skip Barcelona and tour cities closer together to avoid spending time in airports.

Go with only 4 cities as others suggested. No matter how you cut it you will be travelling at least half a day every time you move (packing , checking in and out, travel time, unpacking ). Your flight might be 1 hour long but you still have to get to and from the airport and wait to depart. All those cities offer great opportunities for day trips which can add variety to your stay. Have fun.

What does "seeing a city" mean to you? In your plan you get two full days in each; if all you want is a quick glimpse of a few major highlights, it's perfectly doable. However, you might well end up feeling you've just seen a succession of postcards; you really won't have time to get a feel for any of them, especially given that each is in a different country with very different languages/culture. And don't underestimate how tiring it is to to re-adapt to a new country each time, in your program, you barely have time to work out how to get a bus ticket before it'll be time to move on to a different system.

Also, out of 17 full days in Europe, you have five given over to travelling- that's nearly a third of your time in transit. Fine if you enjoy the travelling part - for me taking trains or buses through the country and seeing the landscape is part of the fun - but your post sounds as if you just want to get that part over to reach the next place. You won't get much else done on travelling days; flying is theoretically quicker, but as has been said upthread you won't save much time once you've factored in getting to the airport, check-in time and so on... Trains will get you town center to town center, so you need to check out just how long you really need to get to and from each airport and the price, and how early you have to be at check-in, before you can really compare which is the most efficient. Personally I would fly to Rome and Barcelona, maybe Munich out of your list. London and Amsterdam are both about 3 hours from Paris by train, so faster than flying, also easier to get convenient departure times.

The rest is personal taste, what do you really want to see in each place? Famous monuments, architecture, history, art, food? You could spend a month in some of your destinations without getting bored, depending on your interests.
I would just advise you not to get caught up in the "I've come all this way so I must see everything" syndrome: you won't see everything whatever you do. There are literaly hundreds of worthwhile places that you won't see just in the countries on your list. In leaving one place you give up on the other things you could have found there, if you stay you pass on somewhere else - you'll see the canals in Amsterdam, but not the catacombs in Paris, or vica-versa, so there's no point in stressing over missing out on something. Just make sure that you travel the way you enjoy travelling!