Asking about invitation/desire in basic tense, vs. specific patterns?

Question

When it comes to asking questions, specifically about invitation and want/desire, I've come across sentences that avoid using specific patterns (ませんか/ましょうか, or たい/ほしい), and just use basic future tense instead. When and why can this be done? What are the differences in nuance?

Example 1: 「行くか?」 for "Wanna go?" What are the differences with more "textbook" invitation patterns: 「行かない?・行きませんか。」「行こうか・行きましょうか。」

Example 2: Since politely asking about someone's desires directly is tricky in Japanese, one natural way to ask the equivalent of 「何か飲みたいですか?」 is 「何か飲みますか?」 Does this hold true for all questions about desire? For example, let's say two people on a trip are loosely planning their schedule: Is it OK to say 「明日早く起きますか?」 for "Do you want to wake up early tomorrow?"

Answer 1

"行かない?" is an invitation ("Why don't you go?"), so the speaker expects the listener will go. On the other hand, "行く(か)?" ("Wanna go?" or "Do you go?") is a neutral question about the listener's intention, so the listener can respond with "No" without disappointing anyone. "行こうか" is like "So let's go" or "It's time to go", and it sounds like the speaker isn't even asking about the listener's intention (maybe the speaker already knows the answer).

"明日早く起きますか?" has to be translated with or without want, depending on the context. If they already have a detailed schedule for tomorrow, then want is unnecessary because the question is to confirm a schedule that has already been set. However, if this is a question for deciding what to do tomorrow, it is natural to say "Do you want...".