I agree that some of these overlap. As noted in the instructions, there may be more than one possible answer. So how do you know which is the correct one to use? In many cases, it's just about set phrases or certain verbs going with certain nouns or ideas.
For example, in #4, when you make old paintings look new again, it is called "restoring" them. The field is known as "art restoration".
And when you make a building look new again through some restorative construction, you are "renovating" it. So in #8, we can say the the hotel is being renovated.
When a person is made to look or feel younger, we say they are "rejuvenated". So in #6, his job seems to have rejuvenated him.
When we perform a situation or an event as it happened in the past, we are recreating it. So, in #3, the play recreated the events of the past.
When you rebuild an area in a city or town, we talk about redevelopment. So in #5, the money is used to redevelop the commercial area.
When someone loses something and must get it back, for example their health or wealth, we talk about regaining it. So in #2, it took him a long time to regain his health.
A very specific verb to describe being put back in a position you once held is "reinstate". It can also be used to say that a law/rule/order is being put back into place. So that's why in #1, she was reinstated as manager.
And that leaves "regenerate" which is what a starfish does if it loses a leg: it grows a new one. But you can also use "regenerate" to restore something to its original strength or to change something radically for the better. It is often used to talk about cities, so it fits with #7: There are plans to regenerate the inner-city area. (You could also use "redevelop" for #7.)