The meaning of 是...的 in 有什么事是我该做的吗
That's simply a weird sentence. 

In fact your simplified sentence 有什么事我该做吗 （是and的 removed) is much more concise and fits better the desired way of speaking Chinese. The only problem is that conventionally attributive clause should precede the noun in Chinese. 

So 有什么我该做的事吗 is more native (的here is just a connection word between clause and noun). Furthermore 事 (thing) doesn't mean any meaningful thing here, you can essentially just omit it and say 有什么我该做的吗？.

Maybe you have seen, the complication of 是 and 的 arises because you want to keep the same word order as in English (attributive clause after the noun).

In fact, an even more native way is to try to never use clause in Chinese. 


  我该做些什么


it looks similar to "What should I do?" but adding 些 implies "something" and makes it mean the exact same thing as "is there anything I should do?".