He's just to going to sell our information and me and my dad are completely screwed.
This is actually non-standard use of pronoun. The standard English sentence is -
He's just to going to sell our information and I and my dad are completely screwed.
In coordination structure, to determine the case of a pronoun is not that difficult. Omit the coordination, and replace it with a single pronoun. Now in coordination also the case of the pronoun will be according to the case of the substituted single pronoun. For example -
Tina and me sat by the window looking down on all the twinkling lights.
Here in the sentence the coordination structure is - Tina and me. Now we are not very sure whether we will use me or I in coordination structure.
Replace the whole coordination with a single pronoun. Let's use we there, and the sentence will look like this -
We sat by the window looking down on all the twinkling lights.
Here the pronoun is in the nominative case, so the pronoun in the coordination will also be in nominative case. So the correct standard English will be -
Tina and I sat by the window looking down on all the twinkling lights.
Now consider the following sentences -
1. Peter and me are going to represent the workers of the coal mine.
2. She and us are going to be good friend.
3. Me and Mike might compete for the trophy.
4. Him and me fixed up the wagon while the others went to town.
All the four sentences above are the example of non-standard English. In all these sentences nominative case would be appropriate.
In sentences #1 and #2, accusative case appears after the coordinator. The coordinators here are treated much like a preposition for case assignment purpose. In sentence #3 and #4 the accusative form is used irrespective of position of the pronoun in a coordination structure.
The accusative use of pronoun in those four sentences are non-standard and are strongly stigmatized, especially the pattern in sentence #3 and #4.