¿Cuál es la diferencia entre caso omiso y caso contrario y acto seguido ?Puedes dar oraciones como ejemplo.

"caso contrario" = "otherwise"
"If there are apples left i will eat them, otherwise, I will have to buy"="Si quedan manzanas me las comeré, en caso contrario, tendré que ir a comprar"

this expression is often used but not a lot though. Others like "Sino" or "De lo contrario" are more common to be heard.
it is not informal at all.

"acto seguido" = "then" when used to introduce an action that happened later like this:
"I bought some food then I realized it wasn't enough"="Compré comida y acto seguido me di cuenta de que no era suficiente"

Again, it is common but not too common. It is not informal at all either.

"Hacer caso omiso"= "Ignore/ Don't pay attention"

"Le hizo caso omiso a la carta que le envié"="She ignored the letter I sent her"

This isn't common and "No hacer caso" or simply "Ignorar" are used instead.
It is not informal at all either.