I was trying to find the time to test your itinerary against the time table, but Annika already did it.
She also said it more gracefully than I do, but my overall impression is that I agree completely with your comment:
<<"I feel the above plan is not smooth, and not so practical">>
Basically, you are trying to do too much in the limited time available for your trip. It could interfere with your goals. Annika has made some good suggestions on how to refine your plan. You will have better experience of Switzerland if you spend more time in each of fewer places, or spend more time in the Interlaken region, or pick a base with more variety in the things to do nearby. More on that later.
That could mean skipping Northeast Switzerland. By the way, for those seriously interested in culture and history, I'd pick the Abbey and old town of St. Gallen before the Rhine Falls. Just my opinion. it is further than the falls from Zürich, Luzern, or Interlaken, however.
Since it is very clear from your posts that you have spent some serious time studying what is interesting to see and do in Switzerland, perhaps you also know this trick, which I will explain for the Golden Pass route as mentioned by Annika:
If you tell the timetable to find trains from Interlaken to Montreux it will find the fastest routes (which go through Bern). It defaults to the fastest route. To find the Golden Pass routes, use the same start and finish, but enter Zweisimmen in the "Connection's" or "Via" box. You will see different descriptive terms for trains at different times. They are explained in the link for the Golden Pass provided by Annika.
Interlaken is a god base for your longer excursions, and some of the Jungfrau region will be acccesible (and beautuful, if the weather is clear) even though many services and facilities will be closed for between-seasons vacations and maintenance. (For example, as Annika noted, the timetable would show you that the cableway from Grindelwald to First is not running then). If possible, if you wish to see the Jungfrau region, it is a good idea to stay 2 or 3 night close by, as you are doing, to increase your chances of finding a clear day. It is no fun to go up to the top by the expensive cograil ( or even partway to Kleine Scheidegg) and be inside a cloud. Kind of boring. ;-(
As Annika said, it is a good region to focus on.
If you want to see that region, a better travel time is in late June (after the 15th to allow the snow to go away on the higher trails or in September and usually, very early October, before it snows much at high altitudes.)
Also, trips to Bern, Luzern, Zermatt, and Montreux are all reasonable day trips from Interlaken.
Each has a lot o see and do:
although I think you may have already looked at that information?
If you do consider changing your itinerary to permit a visit to Val Verzasca, it would be possible to do it in a long day from Luzern. Once you are at Sonogno, there is not a lot to spend time on, although the famous bridge partway into the valley at Lavertezzo is quite attractive. It is not a short walk from either end of the valley, so , if you go in by bus, there might be some scheduling concerns.
So, unless you decide to do a serious hike up the mountainsides, you will not spend as much time there as you might in one of the cities discussed. The photography is pretty good, in part because the extensive use of stone in all forms of construction provides nice textures.( Sorry for the misspelled name in the attached file.)
Consideration of a map of Switzerland, in conjunction with the time table, can help you think about a good base. Interlaken works for much of your goals.
Luzern is another place to consider. The variety of tourist attractions and natural beauty, although less awe inspiring than the Bernese Oberland includes a particularly beautiful lake, nearby "gentle' mountains (the Rigi) and high ones (Pilatus, and, a bit further, Titlis at Engelberg.) The old town is very nice, and there are some world class museums, especially the Rosengart Collection close to the main station.
Luzern has the advantage of a lower altitude, so that it will be well into the first part of Spring in late April. Fewer tourists than later in the season.
I've attached a screen grab from the timetable for a trip from Luzern to Sonogno and return.
3 1/2 to 4 hours, one way.
The time table will not show next April (to far in advance) but the schedule will be similar then as it is now, so I use May 1, 2016 for my example.A very long day. Even though the train ride over the Gotthard Pass is worth a detour, as they see, to travel on, I'd still not think time spent going to the Val Verzasca on a short, first time trip would be a good use of time. Personal opinion, of course. We enjoyed it when we went there while staying in Locarno for a few days. But,there is a lot more "Switzerland" to experience in the central part of the country.
Are you hikers?