You do need a database if you want the ability to log users out server side. This is usually done through a second refresh token.

In a single page application it is necessary to access the JWT with JavaScript. Thats why it is so common to save it in the code directly or in the local storage. It is dangerous though, since a XSS vulnerability can be used to access the JWT. This would be totally different with a cookie that is stored with HttpOnly.

No, why? It is very often not necessary to make this accessible to JavaScript, except you are working with refresh tokens. But this is mostly not necessary and overused.