THATS NOT TRUE! were ahmericah! fuck yea! we aint got no monarchs! Our president does not have the power to send troops unofficially into other countries without declaring war or spread propaganda to change public opinion into supporting an invasion of a country.

It's entirely possible. Most non-governing monarchies still have themselves inserted into a position of power and simply rubber stamp the actions of the governing bodies. In the UK the Queen (or future King) has the ability to Veto a piece of legislation. Similarly in Canada, Australia and New Zealand the Governor General holds the veto amongst the other reserve powers.

However, the Japanese Emperor has no 'reserve' powers. So its ability to take power may be more restricted, however there's always a good chance that they would be restored to power in the event of an uprising.

The chain of succession has been broken a few times, most prominently by assassination, where the next candidate assuming the throne had, ahem, well, ample reasons to be glad they hadn't invented DNA testing yet. The unbroken succession is largely an article of faith, concocted in the late 1800s in the service of turning an often fractious bunch into a modern nation-state.