Menu is not working and theme is not working for category page

1) Menu is not working.

You should be able to solve this issue by following the instructions here:
https://jetpack.com/support/mobile-theme/#custom-menus

2) Category page still being displayed from desktop theme in mobile instead of Jetpack mobile theme.

Do you use a caching plugin on your site? If so, could you make sure it’s configured to be mobile compatible, and never cached desktop pages to mobile readers?

It should help.If you still experience issues, could you post your site URL here, so I can have a look?

If you want it to remain private, you can also contact us via this contact form:
https://jetpack.com/contact-support/

I am using wp super cache and I think I have to use one or other caching system to load page faster.

Here is my website link : https://news.worldsnap.com/

Could you then try the following, to make sure WP Super Cache is configured properly on mobile:

1. Go to Settings > WP Super Cache > Advanced
2. Check the following option: “Mobile device support. (External plugin or theme required. See the FAQ for further details.)”
3. Save changes
4. Go to Settings > WP Super Cache > Plugins
5. Enable the Jetpack Mobile Theme option there
6. Save your changes
7. Go to Settings > WP Super Cache > Contents
8. Delete all your cache

Let me know if it helps.

This settings “Mobile device support. (External plugin or theme required. See the FAQ for further details.)” is already checked.

Have you tried enabling the other setting as well?

Let me know how it goes.