Late April / early May in Switzerland & the Alps

It is certainly between seasons.

You will need to look at the timetables to see which cable cars and trains are closed for their annual maintenance.

Most road passes will still be closed due to snow.

For the Schilthornbahn you cannot drive any further up than Lauterbrunnen or Steckelberg. It is then cable cars up, if they are running.

Weather has a high probability of overcast or heavy at that time of the year.

That time of the year is when the snow starts to pull back at lower altitudes so the farmers will take advantage of all the manure they have been stockpiling all winter, turn it into soup and spray it on the fields to fertilise them. Brown fields is not necessarily due to dead plants or bare soil from the winter. You'll smell it before you step in it.

Some hotels will be open, certainly in Lauterbrunnen and especially in Interlaken. Check before you travel.

Will high-altitude sights, hotels and such be closed either due to weather, it being an in-between season or both?

Both. The high-altitude resorts go on hiatus because it's too warm to ski but too wet, muddy and unsafe to hike. I'm pretty sure that Piz Gloria is closed, but check the schedule.

As Nigel noted, the "high altitude views", if the lifts are even open (most, if not all are not) are more likely to be a dense white out than anything else. The "villages" are not at their best either because most of the hospitality infrastructure is closed and the farmers use this lull in the tourist traffic to fertilize the pastures... which consists of liquified manure sprayed through a power hose. It smells... um, ripe...

Unfortunately, this is the absolute worst time of year to visit the Alps.

Thank you both. Between your responses and more that we've read, we're going to reschedule this to September 2016. That sounds like one of the better times of year to go to this region.

I think overall the best time for the Swiss Alps (non ski season) is the first half of June.
http://www.alpinetraveler.com/whentogo.html

good decision

We were in Lauterbrunnen in July one year and did a few easy hikes, we were told that on some paths the snow had not melted till June!

June is not completely safe either. If the previous winter and spring are particularly cold, significant amounts of snow can linger on some of the high elevation hiking trails.

For hiking, September is probably the best time. The weather tends to be relatively dry ("relatively" being a key word), temperatures are comfortable, and the snow pack is long gone.

OP mentioned "high altitude sights". Hiking of course requires close attention to conditions at all times.

We used to travel at that time of year due to kids school vacation. We skied under the Matterhorn and took the tram up to the Schilthorn in late April. We stayed in Zermatt for 3 days before we were able to actual see anything above the elevation of Zermatt. On day 2 we went to do our day of skiing, the guy at the rental shop recommended we wait a day because they were calling for a beautiful day that following day. It was so worth the wait! We had a beautiful, sunny, blue sky, spring skiing day! While up there we also went into the Ice Palace.
Trummelbach Falls was another stop we did near the Schilthorn. Had lunch in Murren. We spent 2 days in Grindelwald also. It was much too early for hiking in the higher elevations but the weather was nice for the most part.
From what I've read, August to mid Sept is the best for hiking in the high elevations.
We had no reservations anywhere and had no problem finding 2 rooms on the spot.
I would choose a better time of year if I'd had a choice since we like to hike but it was a very nice trip.