What is the role of 给 in this 把 ... 给 V construction?
There are two kinds of usage of 给。Let's use the three sentences A, B and C as example:

风把门给吹开了。(A)
门被风给吹开了。(B)
门给风吹开了。(C)

In the sentence C, 给 means "passive", 门给风吹开了 = 门被风吹开了。

For the two sentences A and B, 给 is used to highlight some emotion a little. If you remove the 给 in the two sentences, they are still correct but they will become the same thing with the sentence A.

For example, your mom told you to close the door, you did but the door was opened because of the wind. When your mom asked you: why didn't you close the door, you can say: 风把门给吹开了。With this sentence, you are telling your mom that it's not your fault.

Another example. Your friend is passing you some paper outside while the wind is blowing, he may say: 小心点，别让风把纸给吹跑了。It means that your friend is trying to warn you: be careful when you hold the paper, because the wind may blow it away.

As mentioned, 给 can highlight some emotion but just a little bit. So normally it's still fine if you remove it: 小心点，别让风把纸吹跑了。--- Totally correct.