1. Is there a way I can add search icon next to primary menu for doing a full site search ?
I couldn’t in customisation.
You could add a search widget to any of your footer widget areas:
https://en.support.wordpress.com/widgets/search-widget/
You could then add a link to your menu to bring your visitors to the widget further down the page, and even an icon next to the menu item if you wanted. If you need help with that, you’ll need to first add the widget, and upload a search icon to your media library that we can use.
2. there seems to be a problem with menu dropdowns. On hovering, the parents used to display childs like a nested branch. It was good with the previous theme. can anyone help help?
I’m not sure what you’re referring to – when I look at the dropdown menus on your site, they work as I’d expect them to.
Is the previous theme you’re referring to Heart & Style? I took a look at the demo here – https://heartandstyledemo.wordpress.com/ – and the dropdowns are quite similar to what I see on your site running Photo Blog, so any additional details you can give me about how you’d like your menus look would be helpful. Thanks.