Category not displaying in my widget

I am making effort to customize recent post default widget to another widget. I add some functionalities to it to achieve what i want. I have successfully done it. I added show category to it but when i test run the widget, it is not showing the category ID even though i tick the check box.
Please, kindly help me look at the code itself and let me know what i have done wrong in the first place? I also wish it to have excerpt text of each post> Must i equally input an $instance element in the widget code or just go ahead to insert excerpt to the main body of my widget?

what i want to achieve is exactly the same as this link.https://bootsnipp.com/snippets/1d3X9. At the end of the whole process, i will create a carousel slide show from it but just want to get everything in order before i implement the slideshow.

Also, when i use theme checker to know the state of my widget code, i keep seeing error message lik https://pastebin.com/Q4vEwtJe

Here is the link https://pastebin.com/aVVt7L6Z of my entire code