Hi jmath, and welcome to MySwissAlps!
Congratulations for your coming honeymoon, and your idea to come to the Alps for that!
I have some experience of hiking on the US East Coast (Shenandoah National Park), but I live in Switzerland. Well, things are a bit different here.
"Huts" have that name from the German "Hütte". In French and Italian, they are called "Refuge". No word, really, exactly describe what it is: a place where climbers and hikers can spend the night safely. Most huts, but not all, have a warden who will tell you where you sleep and will, with his team, prepare the meals. The standard comfort is collective dormitories (approx. 12 sleepers), with mattrasses, pillows and blankets or quilts. You are expected to bring your own silk sleeping bag (for hygiene, not for warmth). Some huts also have four-bed rooms, to get such a one it would be better if you did the trip in September. Otherwise expect snoring etc.
In huts with a warden you can get dinner (usually you cannot choose from a menu) and breakfast. Some, but not all, can prepare sandwiches for you; most would sell cereal bars and chocolate bars, and all provide tea for your bottle if you want. Nowadays most huts have flush toilets, also called WC, but not called bathrooms. You can wash your face and teeth, usually with cold water. In some modern huts, you can take a shower (pay extra for that). It is highly recommended to book in advance. Read carefully their general conditions especially the cancellation rules.
So it's not quite your Shenandoah Lodge, but that will surely be a honeymoon to remember.
Chamonix-Zermatt: probably doable in 10 days. If you take the glacier itinerary you should join a group or hire a guide. If you take the lower route (Alpine Passes Trail, no 6 on this map) you don't need a guide, trails are well blazed. Snow should be expected even in the Summer on higher passes. You need the appropriate equipment, and should train walking 6 hours with a backpack weighing 8 kg or more.
Schynige Platte - Wilderswil: If the above description of huts scares you off, and if you can't do any training, this is for you, It can be done in one day so no hut is required. Snowfall very unlikely. You still need equipment such as mountain boots and a rain jacket no matter what the weather forecast says
Eiger to Zermott: I am not aware of any place called Zermott in Switzerland. If you mean Zermatt, well, it is in quite a different area than Eiger. Eiger, for one, is a peak only to be reached by experienced climbers. You could, in theory, consider trails nos 1, 40 and 6 on this map.
Other suggestion: I hiked last week from the Italian speaking part (Ticino) to the Romansh speaking part (Grisons) in 5 days, on the following route: Airolo - Cadagno Hut (modern hut with showers and 4-bed rooms) - Dötra Hut (small hut, but shower available, 4-bed rooms) - Campo (Blenio) (local guesthouse) - Terri Hut (large dorms, provides sandwiches, no shower) - Vrin (lovely local inn). Trail no 6 on this map.
Sure you don't want to go to the Virgin Islands, or Niagara Falls?