A common way of asking a group of people a yes or no question about something directly related to them is using the pattern:
When I hear this, I immediately jump to an image of a teacher asking a group of students a yes/no question in class and they all raising their hands while saying "はい".
Who liked today's (provided) ...
～だろう forms a question that is mainly addressed to yourself. It's like "I wonder". If you are clearly requesting an answer from someone else, don't use it.
Next, か after だろう in a yes/no type question corresponds to English "whether/if", and it's mandatory. か in a wh-question (i.e., a question including いつ, だれ, 何, なぜ, etc) is purely optional, but is usually ...
えいがはなにをみましょうか implies an invitation. しましょう(か) means "Let's". It means (私達は)何の映画をみようか("What movie should we watch?" Or "Let's watch a movie. What is a good movie to watch?"),
えいがはなにをみますか is a question. It is "What movies do you watch?"
えいがはなにをみませんか is also a question. You are asking someone what movies he doesn't watch. However, it is rarely used and a bit ...
I will answer first your 2nd question, and after that I will address your 1st question.
2. Does the last sentence even make sense?
No, the 3rd sentence does not make sense. Only the 1st and the 2nd sentences make sense:
えいがはなにをみますか。Regarding movies, what will [you/we] watch?
えいがはなにをみましょうか。Regarding movies, shall we [you/we] watch?
Using ません or ...