Feature-Full Widgets with Widget Options
Lots of individual plugins exist to add these features; but if you want to go nuclear, Widget Options adds all these features and a whole lot more. Many of them (such as controlling widget alignment on different devices) I’d personally use CSS to handle, since it’s possible to be more specific and more compatible with existing theme styling that way. But a lot of what it lets you do is cool for sure—particularly the paywall-gated features like “Widget Caching Options.”
In general, if you’re working with widgets and wondering “Why can’t I…?” then Widget Options may be a good choice to check out.