Custom theme based on Twenty Eleven, update necessary? Widgets not working

Dear all,

Our site has a custom theme based on Twenty Eleven.

Since widgets such as amongst others AVH Extended Categories do not work anymore, that is, we literally cannot open the widget and thus cannot add postings to the categories and/or alter information in the footer anymore.
Peter van der Does thinks that this may have to do with a necessary (?) update of the Twenty Eleven theme.

Questions:
1. Is that indeed the solution?

2. If I update the Twenty Eleven theme to 1.6, do I have to alter the whole site again within the custom theme? That really is my fear.(FTP shows under ‘themes’ resp.: PAcustom theme, Twenty Eleven, Twenty Twelve).

3. Can I delete theme Twenty Twelve through FTP without risk? (I do not use it. I deleted it last week whilst making other adjustments as well, and it is back under the themes section.)

https://paonline.nl

Thanks so much in advance. Much appreciated1
Best regards.