Great questions.
You can set your Internet Gateway up to allow connections through using some Port forwarding and some other networking wizardry. I’d advise that it’s not the most secure way of accessing an OctoPrint setup from the wider internet though. For that, you should use one of the Plugins developed for OctoPrint. This one called OctoPrint Anywhere allows you to access your OctoPrint server via a secure login. Instructions & Link here.
In regard to the DLink switch: having a wired network connection between your Raspberry Pi and home network can reduce traffic on the WiFi interface, which might speed up your internet connections on other devices using the same WiFi network. However, it’s really not necessary.
