MSB, you have a very accurate understanding of things. A DoW is certainly not a prerequisite to engage in hostilities, its significance is more one of international law and diplomatic relations, and does not for most purpose have a military significance at all as far as engaging in actual combat operations are concerned. It does enable us to undertake farther-reaching actions in areas like commerce interdiction under international law with fewer consequences, especially where third parties are concerned. For purposes of Presidential powers, though, a 'State of war' or 'In time of war' tend to be more a factual issue than a legal one, and unless the particular power seems to actually require a DoW, de facto war may trigger it. Also there are 20-some different flavors of 'State of emergency' or 'Condition of war or emergency' grants of extraordinary powers to the Executive, in most of which the Executive himself is granted to power to make the necessary finding which invokes the power.
The WPR is of debatable legitimacy as a law, and stopped short of becoming one in an unwritten drug deal between the Legislative and Executive branches, where the former backed off trying to pass it over a probable veto, and the latter agreed to voluntarily follow it rather than either side having to take it to the acid test of the Supreme Court and have one or the other of them lose big. Even as written, it allows the President some pretty-damn-near-complete freedom of action, but for only a limited period; the only consequences are the ultimate power of the purse after that time if the President decides to flout it, but that ultimately requires further Congressional action to actually de-fund the effort specifically or re-program the money he's using.
The Navy has a historical and continuing mission known as "Freedom of navigation exercises," using warships to pass where, to our understanding of the international law of the sea, they are allowed to go. We do this since acquiescing to things like 'Naval exclusion zones' that do NOT appear to us to be supported by customary maritime law tends to legitimize them, and we can't have that. It was such an exercise that was involved in the shoot-down of two Libyan aircraft over the Gulf of Sidra under President Reagan, when Moe the Q somewhat-presumptuously declared a "Line of Death" far outside the normally-accepted international sea boundaries of Libya. This mission is routinely conducted in off-shore waters of potentially-hostile powers and has rules of engagement of its own, which allow the commander on scene to react as required to protect his force, without any screwing around notifying Congress.
Now, if there were to be a planned reprisal or punitive attack rather than just defensive reaction during a naval passage, THAT is something for which a President would generally decide to 'voluntarily' advise Congress in advance under the WPR.