Lucerne or Interlaken?

1. Yes

2. Yes

3. Interlaken too far away and boring. Try Wengen or Lauterbrunnen (Google image them)

4. You can do it when you get there. What if you book and pay now and you get there and it's a week of rain or overcast?

You can stay 2 days in lauterbrunnen which is centrally located in the Berner Oberland. Then, you can do day trips to Jungfrauch and Mt. Titlis. Lucerne is a good base too

Mountain weather is unpredictable, so you need to be flexible, and have multiple opportunities for mountain top visits.
We were in Switzerland fall, 2017, and had a spectacular visit at Mt. Titlis. Even our RS Guide was wowed by the lack of clouds and the visibility. And- there's walking on that suspension bridge....
For the Matterhorn, the clouds cleared just as we arrived in Zermatt.
Our day in Murren was overcast and rainy. We watched the skies all day and hoped for the clouds to clear- no luck at all. The next morning it was an even heavier downpour. So- we didn't experience the Jungfrau views.
From Bern, we had clear, distant views of the Jungfrau- not a cloud in the sky! What I would have given to be mountain top that day!
So mountain weather and precise planning do not always work in coordination.
Safe travels.

A few questions.....

How long have you got for this portion of your trip? What are you most interested in, mountains and incredible scenery or city touring? I wouldn't suggest staying in Interlaken as it's just not the same as the experience in the Lauterbrunnen Valley.

If you only have time for one location, Lucerne makes a good home base as you can easily travel from there to the Lauterbrunnen Valley (Jungfraujoch) and Titlis / Rigi / Pilatus.

One point about visiting the Jungfraujoch, is that it's a good idea to check the webcams to see if the weather is clear. If the top is shrouded in clouds, you won't be able to see much. It's an expensive trip so best not to waste that pricey ticket. I'd suggest going as early as possible. The trip from Lucerne to Kleine Scheidegg (where you connect with the Jungfraubahn) is 3H:13M with two changes. If your primary sightseeing goal is the Jungfraujoch, it would be better to stay in Lauterbrunnen or Mürren in order to reduce the travelling time.

While in that area you could also visit the famous Schilthorn, which was featured in the James Bond movie, On Her Majesty's Secret Service. That's also a bit pricey (although cheaper than the Jungfraujoch), so you may not be able to do both.

Between the two, I’d use Luzern as home base. It’s got endless charm with lots of activities- shops, restaurants, day trips. I found interlaken a good transportation hub but touristy. Murren is spectacular and a short distance from interlaken.

Thank you everyone for your suggestions. I have decided to stay in Lucerne & visit Titlis & Juangfraujoch.

1) I have booked the train ticket till Engelberg. Do I need to book the tickets for cable car also in advance? or Would you advise me to buy it on that day itself on the counter? Do the prices vary, online vs at counter?

2) Do web sites like loco2 or trainline.eu provide tickets for the entire journey of Lucerne-Interlaken-Lauterbrunnen-Kleine Scheidegg-Jungfraujoch? If not, how should I book those tickets?

1. You don't want to book a cable car in advance to the top of Titlis as you can not get your money back if you go up there and can't see anything because of the weather. The prices are fixed
2. No You have to use https://www.sbb.ch/en/home.html

1) I have booked the train ticket till Engelberg. Do I need to book the tickets for cable car also in advance? or Would you advise me to buy it on that day itself on the counter? Do the prices vary, online vs at counter?

Do not "book" tickets in advance for Swiss trains, it is totally unnecessary. There are some discount fares on some less popular times on long distance trains, but generally there is a fixed price. No discount for advance internet purchase. Same price for every train every day throughout the year. You just buy a ticket at the station on the day, it is valid on any train that day. No seat reservations needed or usually possible.
And doubly do not buy in advance for mountain trains, look out the window in the morning first to see what the weather is.

2) Do web sites like loco2 or trainline.eu provide tickets for the entire journey of Lucerne-Interlaken-Lauterbrunnen-Kleine Scheidegg-Jungfraujoch? If not, how should I book those tickets?

These sites are good for many countries, but not for Switzerland. They do not offer all tickets. just look up times and prices on the SBB website ( https://www.sbb.ch/en ), and buy at that price at the station.