native English speaker here. You can say that and it would convey the same meaning more or less, but if you use should it implies that you have more of a choice in the matter. If for example, someone's mother told them to go mow the lawn they would probably ask "do I have to do it right now" because they really have no choice in the matter should on the other hand gives you more flexibility though both questions will get you a yes or no answer
Good question! :)
"Should" means "sollen" or "sollten". "Müssen", is stronger and means "must", "have to" or "need to". The lines may get blured for idiomatic phrases (like "You should know..." which could be "Sie müssen wissen..." or "You should talk!" - "Das musst du gerade sagen!"). Most of the time "should" - "sollen" and "must" - "müssen" have distinct meanings.
"Shall I..." is tricky as you use "shall I..." but "will he?" for future and "Will I..." but "Shall he? " for a more assertive expression. When I tried a translator for "Shall I? it gave me "werden" instead of "sollen", but never "müssen". http://german.about.com/library/weekly/aa032299.htm http://www.merriam-webster.com/dictionary/shall There was a use once upon a time for "shall" that meant "have to" but that is archaic.
For each sentence (or sentence part, for multi-part sentences), one verb gets conjugated and put in either second or first position (for statements and questions without question words, respectively). All other verbs, of which there can be two or more additional, all come last in that sentence (or sentence part). Just the way it is. All modifiers, all times, places, direct and indirect objects, all come before the final string of verbs.
(There's a joke, that if a German starts telling you an involved sentence, and dies before he gets to the end, you're left holding a bag full of nouns with no idea what to do with them.)