Switzerland first time travel - 10-11 days

Hello and welcome to the forum!
It would help if you could share your budget. You say that you are on a very low budget but what does that mean? How much can you spend per day on food, lodging and transportation?
I did a quick search of airbnb rentals in Lucerne and there are quite a few listing under $50 per night for a single traveler. For that price most of them will be private rooms in someone's home or a shared room in a dormitory set up.
If you truly are on a low budget then hostels might be a good choice as well.
Switzerland is one of the most expensive countries in Europe so you should be prepared in advance and plan out your expenses before leaving home. Food and transportation are both quite costly.
Since you have a US passport there are no other documents needed to enter or travel within Switzerland.
The last half of your trip entails a lot of jumping around and 1-night stays. This is a very exhausting itinerary. I would add another night to Lucerne at the very least. Rather than staying in Bern and traveling up to Gimmelwald for the day, why not just stay in Gimmelwald? If you only have two nights in Bern you're using your one day there to go somewhere else, which really doesn't make sense.
Another suggestion might be to just take the train from Geneva to Lausanne - it's direct and very easy.
We have quite a few experts on this forum who I am sure will jump in with specifics about train travel and what might be the most economical choice for you.

My budget for travelling inside Switzerland is $2000 US dollar , which includes
$460 for 15 day train pass,
$75/night for Airbnb - Private Rooms ( I will have 2-3 free nights as I have friend's place in Lausanne, So I might be spending about $700 for accommodation maximum I guess),
$40/day for food so approximately $400 for 10 day food. I want to keep $400-$500 as cash in hand as i am sure I will need it.

I think you are right that I don't have to stay two nights in Bern. So one night in bern then travel to Gimmelwald, stay there for a night and travel to Lucerne after that.

Can you also advise how should I go about travelling to Asher hotel (wasserauen) as it is my childhood dream to go there and I cannot miss that. I will also prefer to spend a night in that hotel. Should I travel to Zurich from Lucerne and then go to Asher hotel and then come back to Zurich to take departing flight, or should I travel directly to Asher hotel at wasserauen from Lucerne and spend night there and then go to Zurich.

Thank you.

A Swiss Travel Pass will fully cover trains, buses, and boats between all the cities you mention, including to Ebenalp for the Berggasthaus Aescher-Wildkirchli. Discounts of 50% apply to some mountain lifts around Lucerne and above Muerren to the top of the Schitlhorn. See the "Area of Validity" map at http://www.swisstravelsystem.com/en/home/support-en/Frequently-Asked-Questions.html?id=137558.

Coverage is pretty comprehensive, but the pass is not cheap, as you already noted. You might also be able to get by with a pass for 8 consecutive days at US $380 per person, which you would start using after leaving your friend.

Most of us here also would choose to sleep in the mountains at Gimmelwald or Muerren instead of in the city of Bern, in order to have more time there and to wake up to a hopefully-fabulous view. Sometimes mountain weather is clearer in the morning. Gimmelwald has a hostel if you're looking for that kind of budget place, as does Lauterbrunnen, in the valley.

Check schedules from Lucerne to Ebenalp on the SBB Swiss site or through the DB German link at How to Look Up Train Schedules and Routes Online. Some departures connect at Zurich and others don't, depending on the time of day.