If Biden wanted to, could he start a conflict against Russia without congressional approval. If not, what approval would he need? If so, what would be the theoretical limitations to his power and military authority?

He could do it for 60 days, at which point to legally continue, Congress would need to authorize an extension. A declaration of war is unnecessary, but an authorization of force (which is let’s call it a more polite euphemism for the same end effect) is at least.

Continued military operations beyond that time would trigger a big political mess. As a practical matter, military forces would still most likely follow presidential orders while the president was either forced to order an orderly withdrawal by Congress, or Congress gave in a retroactively authorized force.

If the political situation was intractable, you’d likely be looking at an impeachment hearing.