>> 15 days starting April 16 ... Zurich, Interlaken (and hopefully Lake Brienz, Lauterbrunnen), Geneva, Lucerne, Montreux, and Bern.
Unless there are specific things you need to do or see in Zürich and Geneva, I would skip them in favour of more beautiful places.
>> My biggest concern is that while we are in Interlaken (4 days), we are in between seasons where we can't ski, nor can we hike. I would love to be able to do some small trails, and at least hit some viewpoints - what would you recommend? I would like to be able to see the Harder Kulm viewpoint - is it open year round? As well, for the Interlaken area, I would like to do some day trips and am wondering what towns would be the best.
Yes, some mountain transport will be closed for maintenance when you are there, or will not yet have recommenced operating after the winter break, but you will find plenty of things to do in 4 days. Valley and lakeside walks at low altitude should be possible, e.g. Lauterbrunnen Valley, Thunersee, Brienzersee. However, in my opinion, lakes do not look their best in winter and very early spring, unless the surrounding area is snow-covered.
The funicular from Interlaken to Harder Kulm is not open all year, but will be operating when you are there, as it reopens on 14 Apr 2018.
Boats on the Brienzersee (Lake Brienz) recommence operations on 30 Mar 2018 after the winter break. Boats on the Thunersee (Lake Thun) will also be operating.
All the following transport/attractions will be operating/open during the period 16-30 April. (You did not tell us where you will be on particular dates.)
- Train to the Jungfraujoch - operating
- Gondola Grindelwald to First - operating
- Cable car Mürren to the Schilthorn - operating until 22 April 2018, then closed for maintenance 23-27 April
- Cable car Lauterbrunnen to Grütschalp - operating until 29 April 2018 inclusive, then closed for maintenance 30 April until 4 May. When it is not operating, you can still get to Mürren on the cable car from Stechelberg.
Interlaken and nearby areas
- Aareschlucht (Aare Gorge) Meiringen - reopens (west entrance only) 30 March 2018
- Ballenberg Open Air Museum - reopens 14 April 2018
- Funicular from Lake Brienz to the Grandhotel Giessbach - reopens 25 March 2018
- Gondola Beatenberg to the Niederhorn - reopens 21 April 2018
- Schloss Spiez (Spiez Castle) - reopens 30 March 2018
- St Beatus Caves - reopen 17 March 2018
- Pilatus - the cogwheel train from Alpnachstad to Pilatus Kulm does not recommence operations until early/mid May, but you can go up and back on the cableway from Kriens.
- Funicular and cable car to the Stanserhorn - reopens 14 April 2018.
- Gondola to the Titlis - operating
- Trains to the Rigi - operating (but cable car between Weggis and Rigi Kaltbad closed 3-27 Apr)
- Territet-Glion Funicular - operating
- Les Avants-Sonloup funicular - operating
- Cogwheel train to Rochers-de-Naye - operating
- Funicular Vevey–Chardonne–Mont-Pèlerin - operating
- Cogwheel train Vevey–Blonay–Les Pléiades - operating
You can find out more about most of these areas and attractions by searching the MySwissAlps website.