Berner Oberland in April 2017

Check in hotel, Option A: walk around interlaken, head up to Harder
Kulm via funicular (is it open in April?), go to Grindelwald for
dinner? - Option B: Head to Grindelwald, hike to First (2.5 hours?),
return to Interlaken walk around main street and grab dinner. Next
morning check out, leave to Bern for a day trip (we plan on leaving
our bags in a locker on the train station in Interlaken) Return to
Interlaken by end of day and head to Murren to check in to our hotel
(this is where I'm not sure it's a good idea.. I already booked the
night on a hotel in Murren but I'm not sure if it's feasible to get
there around 7-8pm after a day trip in Bern) Next morning, check out
hotel (leave bags in hotel?) and head to explore the Schilthorn, then
hike down via the north face trail Walk around Murren, head down to
Lautterbrunnen in the afternoon to see the waterfalls and explore the
valley (perhaps here we can rent some bikes?)

I'm not sure why you plan to check-in and check-out daily in this region. Just pick a spot like Murren or Wengen (not Interlaken) and settle in for the whole visit. Bern, Brienz, Grindlewald, can all be visited easily as day-trips. All that checking-in and out eats up a lot of valuable vacation time.

Please let me know if this is too ambitious, if the hike from
Schilthorn to Murren and from Grindelwald to First is doable at this
time of year

Those trails will almost certainly be closed in early April. Check with your hotel or TI when you arrive to see which trails are open. The high and medium altitude trails will probably be closed, but lower ones may be passable, such as the easy walk from Grutschalp to Murren. This live webcam shows you what it's like at any time. Today there is still lots of snow on the ground in Maennlichen.

Bicycle rental in Murren or Lauterbrunnen is certainly possible most times of year, but in early April the rental shops may not be open since that is the low season. You may want to do some digging on that one.

Take train back to Interlaken and then head towards our next destination: Zermatt.

You should have a back-up plan in the event the weather is poor when you plan to visit Zermatt. There's no point going to Z if the Matterhorn is expected to be shrouded the whole time. One back-up is to head to Lake Geneva. You'll have no trouble finding good accomodations on the fly in April.

Thank you! We wanted to explore Interlaken AND the mountains around too, that's why we decided to stay one night in Interlaken and another one in Murren. We are also doing a day trip to Bern from Interlaken, so it's easier to do it if we stay one night in Interlaken. I've been reading about all this closures due to weather but can't find one calendar or schedule that tells me which trails and which cables/gondolas/funiculars will be closed or open... Is there one website with the schedule/calendar for trails?

The trails open depending on the weather and seasonal snow melt. There is no set schedule. Many of the higher elevation trails are not completely passable until June, although this varies from year to year. And at lower elevations, they can be quite muddy in April...

... and this may also be the time to mention, if you haven't already read about it. As soon as the snow melts, the farmers use this opportunity to fertilize the pastures... with liquified manure. In the usual wet conditions of April, this turns the pastures into a muddy-poopy soup, and much of this can run off into the trails. This is one of the major reasons why much of the infrastructure shuts down. Most people would probably prefer not walking through feces on vacation...

Also, expect lots of overcast and foggy days in April.