This answer already answers:
Are compressor maps such as this one, drawn for one blade row only or the whole compressor?
The shown compressor map is indeed for an entire compressor as can be seen from the pressure ratio.
If it is for one blade row, can you predict the behavior of the next blade row from the compressor map of the previous row?
No you cannot predict the behavior of the next row based on a single stage map, single stage maps do exist though. In theory you can stack these maps to create a complete compressor model, but various loss models prevent that you get an accurate representation. But, since these maps are very hard to get, performance engineers are happy to get these data even if it is a stage map. Also, when modelling gas turbines you may want to split a compressor to model the flows or torques at a certain stage more accurately. We've done this in the past to calculate the individual stage torque contribution of a 3 stage fan of a low bypass turbofan fighter engine.
I am guessing that the blade rows of say the HP compressor have the same speed lines because they are on the same spool, so would they move up or down the speed line?
The smaller the blades the more important losses become, e.g. a constant tip gap in the compressor will have a smaller effect on large compressor blades than it will have on smaller blades. This means that efficiency and pressure rise are lower the further down the compressor stages, on the speed line the operating point would go down the speed line.
If those maps are drawn for the whole compressor, then does that mean that all blade rows have the same operating conditions (in my opinion, they should have different operating conditions)?
As the air is compressed, the inlet conditions for each stage are different. The axial speed is the same though.