Do I need to sand off old finish completely before applying polyurethane?
Generally speaking no, you don't need to completely remove previous finish to apply more of the same finish or a different (compatible) finish1.
But there are some caveats. To ensure good adhesion the surface must be clean, clean clean2, and it MUST be matt.
I sanded one floor with 120 grit sandpaper, and now the finish looks patchy.
If the finish looks patchy now, do I need to sand it all down in order to get a consistent look?
Without photos it's impossible to say, but even with photos we might not be able to tell for sure. It's sometimes very difficult to visually judge what's going on with a surface even when it's right in front of you.
You can try wiping a section of the floor with a rag dampened in mineral spirits or paint thinner and see if, while it is wet, the colour becomes more consistent between the lighter and darker areas. This trick gives a preview of how a surface looks with finish on it, although it's not an exact match it does give a useful idea.
If the result of the above test is positive then you're in luck and a fresh coat of finish should even out the colour. If it doesn't then you're really looking at sanding back further, likely all the way back to bare wood to begin again from scratch — with floor finishing this would often be the normal thing to do anyway.
Side note: in general if you're just looking to refresh a finish 120 paper is too aggressive, especially with certain power sanders. A finer-grit screen would have been preferable here, or a scuffing pad if feasible for the area being worked on and the style of sander.
For the rooms that haven't been sanded as much and retains a consistent finished look, can I apply polyurethane directly without sanding it down further?
As long as their surfaces are clean and matt, yes.
Remember that despite the theoretical yes it's still advisable to do a test in an inconspicuous spot to see how well the poly takes to the previous surface.
1 Many disparate finishes are in fact compatible, although testing should always be done to ensure this even when there's the belief the previous and new finishes are 100% simpatico.
2 Traces of dust, grime, polish (especially those containing silicone), waxes and cleaning agents will interfere with finish bonding and lead to poor adhesion, up to and including delamination (peeling).