Besides being Google's preferred structured data markup format for a reason, JSON-LD is described as "a lightweight Linked Data format" on its documentation site.
And according to this post, "It does not affect the performance of the page because it can be loaded asynchronously." Basically, you can insert it almost anywhere on the page - head and footer are both fair game.
Think of it this way. JSON-LD structured data format describes the content on a page. It is not a script that does a lot of heavy lifting or transforms anything on a page. It is therefore very lightweight, and unless there is another underlying issue, it won't slow down your website.
As for minification of code, yes, that is recommended regardless.