Yes, you can.
You should go with us tomorrow. It might be a good movie and bring you a happy movie night.
First, I would change "a happy movie night" because it is not a natural English expression. You could say something like the following:
It might be a good movie and bring you happiness.
But probably the most natural way to say this is:
It might be a good movie and make you happy.
Cause: a good movie; Effect/result: make you happy
What and can do is express a sequential relationship. Do one thing first, then do/experience another thing second.
I want to eat lunch and go shopping.
This does not have to mean that the person wants to eat lunch first. But it can mean this. It depends on the context. If it is time to eat, then it probably means a sequential relationship.
Clean your room and go to bed.
is almost certain sequential.
Since a cause and effect is also sequential, that is, the cause comes first and the effect second; you can use and to express a cause and effect relationship.
Try it and see what happens
expresses a cause/effect relationship.
Eat your food and you will feel better.
expresses a cause/effect relationship.
Come dancing and you will cheer up
expresses a causal or resultative relationship.
Quit smoking and you will live longer.
expresses a causal or resultative relationship.
They restaurant might have your favorite food and you will feel better.
expresses a causal or resultative relationship.
Open the envelope; it may contain our tickets and we will be able to leave!
expresses a causal or resultative relationship.
Take two aspirin and you'll feel better in the morning.
expresses a causal or resultative relationship.
However and does not always do so:
Take two aspirin and call me in the morning.
does not express a causal relationship. This is expressing only a sequential relationship. First, so this. Second do this.
I'm a native speaker.
EDITED to ADD some examples from the Oxford English Dictionary:
And
8. Introducing a consequence.
a. Introducing the historical sequel or consequence of a fact.
The police chief evidently knew all about him and the conversation was short. (1954 G. Vidal Messiah ii. i. 42 )
He cashed the bottles and I got my twenty cents. (1966 L. Bruce How to talk Dirty i. 17)
His education was disrupted by illness and he dropped out of university. (2004 R. Tames Robert Adam 5 )
b. Introducing the predicted consequence or fulfilment of a command, or of a hypothesis put imperatively, or elliptically.
Spray with Sanfect and you're safe. (1933 D. L. Sayers Murder must Advertise iv. 72 )
Drive out nature with a pitchfork...and she will still come back. Shut up a beaver in the Zoo, and it will still make dams. (1946 R. A. Knox Retreat for Priests vii. 69)
Bloomer...is on probation: one more mishap and he will be out. (2004 Management Today Dec. 27/1)