How is it possible to make ‘Delete’ (I will rename it ‘Replace’) button emulate Browse button?
I mean when hitting ‘Delete’ to actually browse for another file?

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You can use JQuery, see the jsfiddle http://jsfiddle.net/AF2c5/1 for a working example:
The HTML markup:
And the javascript:
You can also remove the original file input and only browse with a custom button: http://jsfiddle.net/AF2c5/2/:
Be aware that hiding the input can break in certain (mobile) browsers. Then you are better off with positioning the original input off screen instead of hiding it, see http://jsfiddle.net/AF2c5/3/:
Hope it helps.