Copy Kentico 11 widget to a Kentico 12 site

Assuming your v11 instance is Portal Engine and your v12 instance is MVC, you can't directly export/import as the widgets aren't compatible with one another. You'll have to manually create your widget in v12 as per the documentation.

can I see the styles or the layout for any of the widgets somewhere?

Again, assuming your v11 instance is a Portal Engine instance, you have to go to the underlying web part to get that layout as widgets are inherited from a web part.

The layout and CSS tabs on the webpart are empty. When I check the .ascx files for the webpart, I can't find where the layout would be. Is there a specific location where the layout and/or the CSS would be defined at? I am looking in the CMSWebPart folder.

In v11, go to the Widgets application and find the widget in question. On the General tab of that widget, look to see what web part the widget is "based on". Next, go to the Web Parts application and find that web part. The web part will have a location in the file system where you can find the code/ The starting path is typically /CMSWebparts.

If it is a base Kentico web part, I'd highly suggest NOT modifying the base web part and/or layout and looking further to find where those properties may be set. You should report back with what web part the widget is based on and we can troubleshoot further from there.