What exactly needs to happen in order for these two countries to be at war? Is it enough that one of the two sides make a declaration of war?
Actual hostilities suffice to conclude that there is a war. Per Art. 2 of the Geneva Convention 1949,
the present Convention shall apply to all cases of declared war or of any other armed conflict which may arise between two or more of the High Contracting Parties, even if the state of war is not recognized by one of them. The Convention shall also apply to all cases of partial or total occupation of the territory of a High Contracting Party, even if the said occupation meets with no armed resistance.
No. No War has been declared, but that's not new, declaring war(that is a specific legal act under international law, that can only be formally concluded by a peace treaty) is old fashioned and neither Russia(as the legal successor of the USSR) nor the US have declared war since in 1941 (EDIT: forgot about the Soviet declaration of war on Japan in 1945 when I wrote that) and if you look at the history of declarations of war since 1945, the last time any country formally declared war on another was Iraq vs Iran in 1980. There have been declarations of a state of war, or of the existence of a state of war since then, but those are a step down).
Yes, but in a weird way. Ukraine is still transiting record amounts of Russian gas through its territory for European customers and getting paid by Russia to do so while fighting it. I think that conceptions of what war is in the 21st century would look very strange to our ancestors.
The present war between Ukraine and Russia started in 2014 and not in 2022. It just intensified in 2022.
The definition of "war" is pretty much elastic, but one can use few common criteria:
- Military personnel involved (checked, both sides, denied by Russia until it became pointless to deny).
- Military equipment involved (checked, both sides, denied by Russia until it became pointless to deny).
- Military hierarchy involved (checked, both sides, denied by Russia until it became pointless to deny).
- Military personnel casualities (checked, both sides, a significant number that cannot be chalked off as "incident").
- Prisoners of war taken (checked, both sides, even small number of them exchanged).
- Civilian casualities (checked, a significant number that cannot be chalked off as "incident").
- Borders moved de facto (checked).
- Borders moved de jure (disputed, mostly unrecognized by other countries).
- Being called "war" in everyday speech (checked).
- Being called "war" in official sources of both sides (not until 2022, from 2022 on - only by Ukraine).
- Being called "war" in third-parties international relations (checked).
- A war being declared (neither side).
Too much "quack like a duck" points.
Under traditional (pre-1945) international law, invasion of one country by another creates a legal state of war whether it is declared or not. Traditionally it was usual for the invaded nation to declare war on the invader, but this was not required for a state of war to exist.
Other answers are correct that formal declarations of war have been rare since the end of WWII and the wide acceptance of the UN Charter. But acts of war may create a state of war even thought contrary to the UN Charter, and even though no declaration has been issued. An actual military invasion is one of the most classic and definitive acts of war, and an invasion that results in armed conflict creates a legal state of war.