Is this similar to Wine which creates a different setting for apps, in its case a Windows environment?
Would using Distrobox create the icons in the menu (searchable) as Wine does and would I have issues with the programs running? Wine programs sometimes lag on my system due having two environments at the same time.
Well, the stuff in your menu are usually .desktop files. You can in theory autogenerate them via scripts.
There are tools also like I think Gear Lever though I have not tried it
No, WINE is a bit different. As I said, Flatpak is closest thing that most people are familiar with that I can compare it to. Though if flatpak is an entire prebuilt package manager, Distrobox is more of a barebone container manager for running terminal apps. While you may have lower performance inside containers, it shouldn’t be by that much. At least not any more different than running any flatpak application.
Sorry, forgot to add that yes Distrobox has an export option that lets you add it to the launcher menu