As regards the reaction of China and Russia, I don't see how a formal "declaration of war" by the US Congress would matter much one way or the other as long as it was limited to war with North Korea...the US military strategy would be about the same. The USSR and China would react to the military reality - as China eventually did - rather than a paper technicality.
A US declaration of war would possibly complicate the diplomatic issue. The US was much better served by having the UN Security Council (in the USSR's ambassador's absence) vote to authorize the war as a police action against aggression. Thus, it was not a US war with various allies fighting alongside the US, but a UN action in which the US was only one of many members (albeit the largest and most dominant) participating.
I am also not sure if a declaration of war would have been a sure thing. The attack was not a direct assault on the US. Probably one would pass, but it would not be unanimous.
I suppose if the US had declared war against the North Korea this would have set (or rather continued with) the basic precedent that a US President needed to ask Congress for a formal declaration before entering a large-scale conflict. This would still not affect the many minor military actions (misc overthrows in latin america, covert ops, in-combat advice to allies fighting insurgencies) the US undertook, and probably not Vietnam, since US participation sort of gradually became a "war" between the US and North Vietnam when no one was looking - but I could see the possibility of formal declarations of war being asked in the following situations: (1)Iran when the US embassy was stormed and hostages taken, (2) Iraq, after the invasion of Kuwait, (3) Afghanistan after 9/11. I suspect # 1 and #3 would have found widespread support in Congress