12 days in Europe

You mean, you want to see Aurora Borealis, the northern light? If you want to see them for sure, you have to go to northern Norway, Iceland, Finland or northern Russia. And stay there for weeks. They are connected to solar storms, which are frequent but unpredictable. It's not something you can do on an all-Europe tour.

This plus Paris plus Swiss Alps plus Rome plus another destination will make sure you see nothing of Europe but airports. And, your budget seems much too tight for this, too.

Any idea if I choose Iceland for 4 days , and my flight should be start and end in amsterdam , I have to go Paris too .

Today is almost September -- do you already have your plane tickets to and from Amsterdam? Train tickets booked far in advance can be cheap, but prices go up as you get closer to departure and the cheaper tickets sell out. FYI, adding a trip into Switzerland plus the €200 cost of getting to the top of the Jungfrau will cut substantially into your $2,000 budget.

The Northern Lights (Aurora Borealis) are not guaranteed at any time, and would be doubtful in October, the best time being Jan-Mar.

Arrive in Amsterdam (2)
Train to Paris (3)
Train to Lauterbrunnen, Switzerland (3)
Train to Rome (3)
Fly to Amsterdam (1)
Depart from Amsterdam

I think your "plan" (such as it is) is not realistic. I would suggest you take a step back, re-evaluate pretty much everything you have said here, and start over with the most basics...
1. How much time do you really have? 12 days? Honestly, 12 days in not very realistic for a trip to Europe for most people (well, it depends on where you are coming from). Sure, you can go to Europe and back in 12 days, and with that you will have a little more than a week of usable time there. With that much time, most people would recommend no more than 2, maybe 3 places.
2. Where do you really want to go? With a bit more than a week, you could see a bit of two places that are easily connected. Of those you mention, a bit of Paris and a bit of Amsterdam probably would work best. Forget "Jungfrauch". Forget Rome. Forget "other east/west Europe city". Forget "Aurora". But don't forget money...
3. Your budget is around US$2000. Assuming that does NOT include your flight ticket to/from Europe, you could make that work for around 8-9 days in Paris and Amsterdam, if you are careful. If that budget includes your air ticket...I don't think that's realistic.
You haven't said where you are coming from, which could have an impact on your costs and time available.

For a twelve day trip I would skip Rome if you plan on being in the north, using various means to get around, including buses to keep expenses down and night trains in Scandinavia I would.