Welcome! 
( I’ve moved your topic to projects so you can get individual answers based on your particular set up.)
- Wemo switches require Wi-Fi. (Wemo bulbs are zigbee.) Zigbee and zwave do not. However–SmartThings itself is still primarily a cloud-based architecture. If either your Internet is down or the SmartThings cloud itself is not available, many things in a SmartThings system will not work. For example, the app on your smart phone requires that the SmartThings cloud be up and your hub have Internet access or you won’t even be able to toggle devices on and off from the app.
(It is technically possible for a phone to use just local Wi-Fi, no Internet, but that is not how SmartThings is designed.)
If you live in an area with frequent Internet outages, it is unlikely that SmartThings will meet your needs. ( also, what country are you in? The available devices does vary somewhat.)
https://support.smartthings.com/hc/en-gb/articles/209979766-Local-processing
2). There are many ways to handle the issue of switches with smart bulbs. See the following FAQ (this is a clickable link)
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you should definitely be able to find smart dimmers that work with dumb dimmable LEDs. But you should not try to combine a smart dimmer switch with a smart bulb unless the switch was specifically designed for that purpose. The two devices confuse each other and you can end up burning out either the bulb or the switch.
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changing your Wi-Fi router should not impact your zigbee or Z wave devices at all. You will need to change the Wi-Fi settings on any Wi-Fi devices, including any WiFi bridge devices that let your Z wave/zigbee devices talk to the Internet. For example, if you have a Phillips hue bridge and some Phillips hue bulbs, you won’t have to do anything with the individual bulbs. You will have to enter the new Wi-Fi password into the hue bridge, but that’s a one step change.
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A number of community members have posted project reports on new House builds that you might find interesting. You can use the quick browse lives in the community – created wiki and look under project reports to find these. There is a list for “whole house” projects and another called “get started”. Both might have topics you’d find interesting.