You can use suppose or supposing before mentioning a possible situation or action. You usually then go on to consider the effects that this situation or action might have.
You can say 'I suppose' or 'I don't suppose' before describing someone's probable thoughts or attitude, when you are impatient or slightly angry with them.
You can use 'do you suppose' to introduce a question when you want someone to give their opinion about something, although you know that they are unlikely to have any more knowledge or information about it than you.