How do resistances and vulnerabilities interact in pokemon with 2 types?

Yes, they cancel out each other. Damage would be neutral.

Yes the damage is neutral.

However, note the special case that if one type is an immunity and one is a weakness, it becomes a single resistance (e.g. electric vs water/ground or ground vs fire/flying)

The battle screen will show a not very effective message (due to rounding error) but the move does neutral damage.

In addition, if the attacking move is super effective against both defender types (such as Electric attacks into Water/Flying Gyarados), the move deals 1.96x damage (compared to 1.4x for a single weakness).

Likewise, if the attacking move is not very effective against both defender types (such as Grass attacks into Dragon/Flying Dragonite), the move deals 0.51x damage (compared to 0.71x for a single resistance). A special case is an attacking move that is not very effective into one defender type while the other defender type is immune to it (such as Fighting attacks into Ghost/Poison Gengar), causing the move to deal 0.36x damage.

Effectively they cancel each other, but due to rounding in the game, the damage modifier isn't quite 1, it's 1,4*0,714=0,9996. This results in the game displaying "not very effective" when you for example attack Dragonite or Salamence with electric attacks. This is very clearly a bug and is hopefully going to get fixed at some point.