"Should I" vs. "Shall I" vs. "Do I" in AE

In colloquial prose, is there some difference to saying "Should I/we", Shall I/we", "Do I/we" to ask someone's advice?

E.g.

Should I call the police?

Sounds like I'm asking someone (or myself) if I would be well advised to call the police.

Should we call the police?

Sounds like I'm asking someone (or a third party) if we would be well advised to call the police.

Shall I/we call the police?

Sounds like I'm asking someone to decide on whether I/we should call the police.

Do I/we call the police?

Sounds like I'm asking someone if it's necessary I/we call the police (e.g. in case something in the neighborhood doesn't look or feel right to me/us)

Am I right on these ones or are there some other differences I didn't notice?

"Shall I?" is an offer. You are poised to take that course of action and are asking if they confirm your decision. It often implies that the speaker is leaning towards the affirmative.

"Do/Should I?" is a request. You are asking what ought to be done. It can be used as above, but does not always indicate the speaker's preferred action and often indicates doubt or ignorance of the proper choice.

For asking advice in American-English in a colloquial context, only "should" sounds good to me.

• There are two problems with "shall." First, as Horatio says, "shall I" sounds more like an offer than a request. Second, "shall" generally sounds affected to me, and I certainly wouldn't consider it "colloquial" in this context.
• "Do I/do we" is a possible way of asking this kind of question, but it just doesn't sound as normal to me in your example sentence. Sorry, I don't know how to explain what general principles cover the usage of do vs. should. Here are some examples to try to illustrate what I mean. "Now what do we do?" sounds OK, for some reason. "Do" also sounds natural to me in certain hypothetical/conditional contexts such as "What do I do if he shoots someone?" especially in the context of asking about the steps of a plan (note that plans are often narrated in simple present like this: "First I go up to the front door, then I knock. You stay behind me and keep alert"). If you're just asking for advice about dealing with an actual situation that's already happened or that's currently happening, "Should I..." sounds better to me.

As for I/we, I'd say the difference is pretty much the same as in other contexts. You'd use "we" if you are representing a group that is asking for advice. You'd also use we if you expect that the person you're talking to is going to be taking action with you. If neither of these circumstances hold, use I.