I’m no expert, but my wife and I have been to Switzerland and Northern Italy—including the Dolomites and the Lakes—six or seven times, so here's my humble opinion:
First, I’ll echo what others have said about the weather in April. Hiking in colder conditions is certainly possible, but in the Dolomites (and much of the Alps more generally) the bigger issue isn’t temperature so much as access: many roads and lifts simply won’t open until late May or even June, climate change notwithstanding.
Another thing to keep in mind is distance. Zurich and the Dolomites aren’t especially close—about 500 km—and even in good conditions that’s the better part of a day of driving. (The train takes about 7 hours.) The nearest Italian lakes—Como and Maggiore—are more like three to four hours away in good weather. If you only have a week, I’d personally save the Dolomites for another trip.
One very appealing alternative is Lake Como.
Zurich isn't my favorite Swiss city, but it does have its charms. It's a nice place to spend half a day or an overnight at one end of your trip. The train from Zurich to Como takes about 3½ hours and passes through some genuinely beautiful Swiss scenery, including stretches of the Alps. Once there, you’ll find excellent low-altitude walking/hiking all around the lake, and the ferries run year-round. Rick’s recommended base of Varenna really is terrific—we’ve taken the train in (from Milan, in our case) and it couldn’t be easier. The town is compact and charming, the waterfront restaurants are excellent, and the ferry network makes getting around the lake effortless.
With Como, you get to see some of the Alps plus the beauty of a northern Italian lake and IMHO it makes for a very nice, active week's trip.
All of this could change if you were able to fly into Venice or Innsbruck—then the Dolomites might be more realistic in a week—but it would still be April, and the same weather and access issues would apply. IMHO, if Zurich is a must, I’d lean into it rather than treat it as an obstacle to overcome.
Day 1 Arrive and overnight in Zurich.
Day 2 Train to Varenna.
Days 3, 4, and 5 Lake Como and environs. 1 day hiking above Varenna. 1 day visit Villa del Balbianello, hike Greenway near Colonno. 1 day visit Bellagio, Como, Villa Carlotta if it's open, etc.
Day 6 Train to Zurich and overnight.
Day 7 Departure
I didn't really answer your questions, but I hope that helps!