Can ‘Related Posts’ will be added to the theme without any plugin?
It’s possible but difficult for an average user. If you don’t want to use the plugin then you’ll have to recreate the functionality. Why make it hard on yourself when you can just use the available free plugins?
I guess, after the recent update, the adsense ads are not correctly fitting into the box
That has nothing to do with the theme. You placed your ads on a Text widget so WordPress added some spaces. Use the correct Custom HTML widget to add your codes.