You will be making your life very complicated trying to merge two widgets if it is even possible depending on the widgets and how they are created. Each will have those own styling and functions making it extremely complicated to do.
If would be much much easier just to code the features that you want from scratch. Is there a particular reason why you want to use multiple Widgets rather than just coding it?
Yes, the reason why is the theme I have. It’s a directory theme, and the search box belongs to the directory. The widgets are part of the theme, and each perform a unique function integral to the theme itself.
Would it really be complicated to say, put a picture on the home page, and make the searcg box widget (that is part of the theme) sit in the middle of the picture?
Without seeing it in action and how the theme is st up its hard to say.
Potentially you could create an area with a background image and place the widget on top but you’d still have a fair bit of playing to do with the CSS and styling to position it and make it look natural.
You might find a plugin or widget somewhere that has that functionality but again without seeing the site it’s impossible to tell how that search is connected to the directory.
Probably your best option is to email the author and see if they can help